I can see why the OP is confused - he's asked what (to him) sounds like a simple question and got many different and often conflicting responses. Many (all ?) people have their preferred technology - some are wedded to combis, some to unvented, I make no secret of quite liking thermal stores (for the OP, I'm purely a DIYer, not in the trade). The only "simple fact" is that there are several options which will all work, and all have their own drawbacks (ie nothing is perfect). It's a matter of deciding which ones will work acceptably, and then which offers the best match for his priorities.
I'll try and explain the options - and their pros/cons. It might not make the decision any easier, but it should help the OP know what people are talking about, and most importantly, allow him to ask the right questions about what he's being offered.
As Dan_Robinson says, a combi is really not a good idea - personally I "very much dislike them", particularly as a landlord where tenants don't like the "no backup when it breaks down" bit.
Once you've discounted instant heating, you are then down to stored heat. You can break that down into systems where you store and heat the water that comes out of the taps (the traditional open vent DHW cylinder, and the unvented DHW cylinder that's being talked about), and system where the heat is transferred to the potable water at time of use (thermal store and heat bank).
The OP already knows about the issues with the open vented DHW cylinder - low pressure.
The other three all suffer from a common limitation - no stored water, so you are reliant on what pressure/flow the mains can provide (with the addendum that you can fit accumulators as mentioned earlier).
All four have the advantage that you can (and normally do) have an immersion heater which will give you a (limited capacity) backup for when the boiler breaks down.
The unvented cylinder works very much like your current one - only it's pressurised (almost) directly from the mains. In terms of draw off rate, you are only limited by your mains capacity until you've used the hot water - and then it goes cold. It then takes a while to reheat - how long "a while" is depends on various factors including the coil capacity and boiler output.
One of the downsides is this pressurisation. They have a very good track record, but that's largely because we have tight regs on storing pressurised hot water. They can only be fitted by registered installers, installation is notifiable, and they should be serviced annually. They can't be installed everywhere because of the need for a safety relief which must be free to vent (in case of over pressure or over heat) somewhere where it won't cause danger.
Thermal stores and heat banks both store heat in water, but this isn't the water that comes out of the taps. The boiler heats the water in the tank (either directly or indirectly via a coil), the DHW is heated by transfer from the stored water. Because there is only a small volume of pressurised water in the coil or heat exchanger, they don't reqire all the safety stuff that comes with an unvented cylinder (they are mostly open vented and hence unpressurised).
In a thermal store the DHW is heated by passing it through a coil inside the cylinder where it absorbs heat through the wall of the tube - which is completely passive and requires no pumps or controls (other than a TMV on the outlet). The size of the coil and temperature of the water set a limit on how much heat is transferred, and so (in common with a combi) there is a set upper limit on drawoff rate though the size of gas supply or boiler don't affect it. For a decent size store like you would be likely to iinstall, I think the mains flow rate will be the limiting factor, not the coil capacity.
With a heat bank, an external plate Heat exchanger (PHE) is used - the PHE can be almost any capacity, and so there need never be a reason for it to limit your DHW drawoff rate. However, the heatbank needs a pumped circuit to pass the stored water through one side of the PHE - so no hot water during power cuts.
With both the thermal store and heat bank, the heating can be taken off the store, which decouples the heating flow rate requirements from the boiler requirements. So you can size the boiler to suit the reheat requirement of the store without having to compromise on it's efficiency when running the heating (or vice versa).
Also, if connected directly rather than indirectly, a well setup thermal store of heat bank can be reheated "top down". The boiler loop is configured so that the water put in at the top of the store is at the desired temperature. So if the store is completely exhausted, some properly hot water is available fairly quickly. Also, during use, the hot water being "used" is also being replenished at full temperature by the boiler - which effectively means the store has a much higher apparent endurance (capacity) for drawoff.
A common disadvantage to both thermal store and heat bank applies if you are in a hard water area. They tend to scale up and need periodic descaling. Scale buildup (less of a problem with the PHE of a heat bank as it a) is a higher capacity to start with, and b) is partially self cleaning) means a reduction in heat transfer rate and so limit drawoff rate and/or temperature.
An unvented cylinder is also bound to suffer some scaling of the coil, but this will only affect reheat times, not drawoff rates.
Lastly, once you mention thermal store or heat bank, many plumbers will either start to foam at the mouth with disapproval, and others will look at you as though you'd started talking Klingon because they haven't a clue how they work. Anecdotally, a common next response is to tell you how rubbish they are and how you need to rip it out as fast as you can.
As to changing the boiler, while a new boiler will (on paper at least) be more efficient, these efficiency figures all assume that it can be kept condensing. In practice, unless you adapt the installation (such as fitting larger rads) to suit, then your old setup probably won't keep it condensing a lot of the time. Weather compensation will help significantly, but won't deal with it completely, while heating. Given the flow rates, I doubt if it'll condense much while reheating the cylinder via an indirect coil.
I'll try and explain the options - and their pros/cons. It might not make the decision any easier, but it should help the OP know what people are talking about, and most importantly, allow him to ask the right questions about what he's being offered.
As Dan_Robinson says, a combi is really not a good idea - personally I "very much dislike them", particularly as a landlord where tenants don't like the "no backup when it breaks down" bit.
Once you've discounted instant heating, you are then down to stored heat. You can break that down into systems where you store and heat the water that comes out of the taps (the traditional open vent DHW cylinder, and the unvented DHW cylinder that's being talked about), and system where the heat is transferred to the potable water at time of use (thermal store and heat bank).
The OP already knows about the issues with the open vented DHW cylinder - low pressure.
The other three all suffer from a common limitation - no stored water, so you are reliant on what pressure/flow the mains can provide (with the addendum that you can fit accumulators as mentioned earlier).
All four have the advantage that you can (and normally do) have an immersion heater which will give you a (limited capacity) backup for when the boiler breaks down.
The unvented cylinder works very much like your current one - only it's pressurised (almost) directly from the mains. In terms of draw off rate, you are only limited by your mains capacity until you've used the hot water - and then it goes cold. It then takes a while to reheat - how long "a while" is depends on various factors including the coil capacity and boiler output.
One of the downsides is this pressurisation. They have a very good track record, but that's largely because we have tight regs on storing pressurised hot water. They can only be fitted by registered installers, installation is notifiable, and they should be serviced annually. They can't be installed everywhere because of the need for a safety relief which must be free to vent (in case of over pressure or over heat) somewhere where it won't cause danger.
Thermal stores and heat banks both store heat in water, but this isn't the water that comes out of the taps. The boiler heats the water in the tank (either directly or indirectly via a coil), the DHW is heated by transfer from the stored water. Because there is only a small volume of pressurised water in the coil or heat exchanger, they don't reqire all the safety stuff that comes with an unvented cylinder (they are mostly open vented and hence unpressurised).
In a thermal store the DHW is heated by passing it through a coil inside the cylinder where it absorbs heat through the wall of the tube - which is completely passive and requires no pumps or controls (other than a TMV on the outlet). The size of the coil and temperature of the water set a limit on how much heat is transferred, and so (in common with a combi) there is a set upper limit on drawoff rate though the size of gas supply or boiler don't affect it. For a decent size store like you would be likely to iinstall, I think the mains flow rate will be the limiting factor, not the coil capacity.
With a heat bank, an external plate Heat exchanger (PHE) is used - the PHE can be almost any capacity, and so there need never be a reason for it to limit your DHW drawoff rate. However, the heatbank needs a pumped circuit to pass the stored water through one side of the PHE - so no hot water during power cuts.
With both the thermal store and heat bank, the heating can be taken off the store, which decouples the heating flow rate requirements from the boiler requirements. So you can size the boiler to suit the reheat requirement of the store without having to compromise on it's efficiency when running the heating (or vice versa).
Also, if connected directly rather than indirectly, a well setup thermal store of heat bank can be reheated "top down". The boiler loop is configured so that the water put in at the top of the store is at the desired temperature. So if the store is completely exhausted, some properly hot water is available fairly quickly. Also, during use, the hot water being "used" is also being replenished at full temperature by the boiler - which effectively means the store has a much higher apparent endurance (capacity) for drawoff.
A common disadvantage to both thermal store and heat bank applies if you are in a hard water area. They tend to scale up and need periodic descaling. Scale buildup (less of a problem with the PHE of a heat bank as it a) is a higher capacity to start with, and b) is partially self cleaning) means a reduction in heat transfer rate and so limit drawoff rate and/or temperature.
An unvented cylinder is also bound to suffer some scaling of the coil, but this will only affect reheat times, not drawoff rates.
Lastly, once you mention thermal store or heat bank, many plumbers will either start to foam at the mouth with disapproval, and others will look at you as though you'd started talking Klingon because they haven't a clue how they work. Anecdotally, a common next response is to tell you how rubbish they are and how you need to rip it out as fast as you can.
As to changing the boiler, while a new boiler will (on paper at least) be more efficient, these efficiency figures all assume that it can be kept condensing. In practice, unless you adapt the installation (such as fitting larger rads) to suit, then your old setup probably won't keep it condensing a lot of the time. Weather compensation will help significantly, but won't deal with it completely, while heating. Given the flow rates, I doubt if it'll condense much while reheating the cylinder via an indirect coil.
