OK. If water is coming forcefully out of the highest rad, then you have probably not got a blockage in the Feed & Expansion pipe.
No reason why you shouldn't have a go at DIY fixing; it may save you the expense of a professional, and will in any case not do any harm if you have to call one in later.
You say some rads are hot and some are cold.
Completely turn off all of the hot rads, and all of the cold except one. Open the valves at both ends (you may need a small spanner to open the lockshield valve) The boiler need not be set very high. Check that the pump is set to high speed. Does this single open rad get hot? If so, you have at least got flow. If not, does it have a Thermostatic Radiator Valve? It might be jammed shut.
Repeat this "all off except one" on all the other cold rads and report back what you find.
Feel the Flow and Return pipes at the boiler, and see how hot the return pipe is. It should be pretty hot, like the one radiator, if there is flow. Feel the pump. It should be about as hot as the pipes. If it is hotter, it is jammed.
If you have some flow, you can at least use a sediment-loosening chemical cleaner like Sentinel X400. I am not a pro and I use this because it is not aggressive or acidic like faster cleaners. You leave it circulating for 4 weeks. You know it is starting to work when the circulating water goes jet black with loosened sediment. Can you do basic DIY plumbing? If so, consider fitting a Magnaclean on a 22mm vertical return pipe. It will cost about £100. As you say parts of your heating system are old, it is likely to have a lot of old sediment in it, and I guarantee you will be amazed and delighted to see how much sediment the Magnaclean traps.
However a chemical clean will not clear a completely blocked pipe, since it can't get down the blocked pipe to reach it.
Also, have a look round all the visible pipes, especially around the boiler, cylinder, pump and 3-port valve and see if you can find a bypass. This bleeds off flow between the Flow and the Return pipes, and may have a valve on it that you can adjust. Sometimes they are hidden under the floor.
You mentioned that some of the downstairs rads will not empty. This is probably because they are on drop loops fed from above, i.e. the pipes to the radiators do not go downwards into the floor. You can drain them by loosening the connections from the rad valves into the rad. It will gush or leak out so empty everything else first, and have plenty of bowls and trays ready, as well as an assistant to empty them. Black radiator water leaves a permanent stain. If you can take up a floorboard some people just let it run out under the floor. Once drained, you can fit a Lockshield Valve with Drain-off which only costs a few pounds,, to each of these radiators. You can put a drain hose on it when necessary.
Post photos of the pipes around the pump, boiler, cylinder, 3-port valve if you can. This may give clues.
You mention that the system has been extended. I hope it has worked correctly in the past since this was done? If not, it might be a design or installation error rather than sediment or a fault.
