yes, it's a combination cylinder and the cold water cistern is there on the top. If you climb up and look into it (there should be a removable lid to keep dirt out) you will find a ball-cock or float valve of some kind in there. It will be a half-inch one, mounted using a large nut. Most likely a brass one like in an old fashioned WC cistern.
A replacement will probably swap straight in. You need a high-pressure, type 2.
It is possible to take them apart and fit a new cone and washer, but this takes longer than fitting new, and a new one is not expensive. If you feel like it, you can clean and service the old one, put it in a clear plastic bag, and put it back in the airing cupboard as a spare for next time, in ten or twenty years.
The colour of the insulating coating on the cylinder is what I'd call "yellow".
Later ones were light green, now light blue.
The installation you have will be very low pressure, so avoid flashy new taps, especially imported ones, and especially mixers with a ceramic cartridge and a single lever or joystick. You probably have traditional individual pillar taps on the bath at the moment, which will fill it in a reasonably short time. You can still buy new, modern taps of similar design, but in my limited experience, they are still smaller than old ones.
It will not give a satisfactory shower.
p.s.
the pipe facing you at the top in the incoming cold supply. I think I can see a white plastic nut and stem, which might be a lower-quality valve.
