You have what is refereed to as a radial, and yes the definitions are confusing, the main distribution board coming into the house can be type tested and called a consumer unit, or it could be before type testing came in and could actually be a fuse board, even if a fuse board it could have the originally fuses replaced with miniature contact breakers (MCB) or have a earth leakage circuit breaker (ELCB) which could be voltage or current operated, and also has a host of other names like RCCB, GFCI and RCD, and the RCD and MCB can be combined as a RCBO. There is a need when people refer to tripping to work out what has tripped, however in the main it really does not matter, 16 amp is 16 amp be it a fuse, MCB or RCBO and it really does not matter if wrong term used when it is quite plain what is being refereed to as it is 16 amp.
There is nothing to stop a modern consumer unit from still having fuses
some times used for electronic equipment as a semi-conductor fuse will rupture faster than a MCB can trip and is required to protect some solid state switches, although never seen them used in a house.
We use a lot less power than we think, my dad fitted a 13 amp fuse for a radial (fused spur) in the kitchen, did not even realise it was there until tried to use two kettles on bonfire night some years back, TV's and the like use very little power, and even if you did use two or three items using the full 13 amp, all it would do is trip the MCB or blow a fuse, so it fails safe. And so easy to reset a MCB not really a problem. With a fuse yes can be a problem as should not use re-wireable fuses in some where under the control of an ordinary person, and finding a shop that stocks a 16 amp cartridge fuse could be a problem.
The only problem adding extra sockets is they need to be RCD protected, and the cable length for earth loop impedance and volt drop should not be exceeded, to be frank with RCD protection even if it was exceeded it would be unlikely to cause a problem.
So yes as long as RCD protected simple add the sockets and leave the 16 amp protective device as it is. If not RCD protected then replace the socket it is coming from with a RCD FCU (fused connection unit) so all after it are RCD protected.
It may seem pedantic, but we go by the design date, not installation date, so if you alter the design, then the alteration has to conform with current regulations, so RCD protected, but the original design does not need altering. However missing RCD protection would still be listed with an EICR (electrical installation condition report) as in the main we don't know the design date, so assume designed yesterday. But would only get a C3 code, which means you can continue using it. Only exception is if some new equipment manufacturers instructions say must be RCD protected.
If we went by the rule book, we would tell nearly every poster you can't do that, but in real terms unless rented property then just do it. You should not break the speed limits, but I bet everyone does at some point, either not seen the sign or still slowing down, hard braking to get speed down on seeing sign is clearly asking for rear end shunt, so we use some common sense, and same with electrics, just do it.
