It looks to me like your motorised valve has only two pipes going into it (an in and an out). The pipe looks like it might be 28mm diameter rather than 22mm.
So unless you have another 2-port valve controlled by the room stat, to turn off the radiators, I think you may have a gravity (convection) feed to the cylinder. This is always slower because the heat from the boiler travels more slowly. It may be worse while the radiators are being heated.
If you can photograph the pipes round the boiler and the pump it may become clearer. Also, feel the two big pipes going into the LH side of the cylinder. One should be "too hot to hold" and the other should be "too hot to hold for long"
This is quite an old design, but the system can be modified without much difficulty.
The valve with the round red head is used to restrict flow to the cylinder so it doesn't take all the heat, leaving the radiators cold. It can be adjusted a bit, but don't do that unless you have a plumber in the house, because when turned, it will probably drip round the spindle. It may also be seized, and could even break. Replacements are readily available.
BTW you really ought to lag all those exposed pipes with Climaflex or similar. If they are 28mm you may have to order it, but 22mm is widely available. Use the thicker grade, where it will fit. It is easily cut with a serrated knife.
If you want to have the system upgraded, wait for the warmer weather, when people lose interest in their heating systems and the plumbers have nothing to do. I would strongly recommend having a system filter fitted at the same time, and doing a chemical clean, because an old system is likely to have a lot of sludge in it. This may remove the need for a later powerflush and reduce the risk of a blockage.
If it was me I would probably have a larger cylinder fitted as well. I can't see if yours has an electric immersion heater, which is handy when the boiler is out of action.
It's easy for me to spend your money. As minimum, let's look at the rest of the pipework. It could still be that the cylinder is scaled up inside, though that's not my first thought.