thanks. Looking at your pics of the LHS, I see a fat pipe near the bottom, and another about half-way up. Those would be the circulating pipes that bring heat from the boiler into a coil sealed inside the cylinder. When your boiler is running on a HW timing, those pipes should be very hot.
I am not familiar with warm-air systems. Feel the pipes, one of them should be "too hot to hold" and one of them will vary between cool and very hot, depending how hot the cylinder is. It will get hotter as the cylinder warms up. the hotter pipe is flow from the boiler, and the cooler one is return going back to the boiler to be reheated. They will not be hot enough to burn you, more like touching a teapot.
the yellow insulation shows it is an older cylinder, and might take an hour or so to heat up. Longer in winter when the incoming water supply is colder. Do you have a timer that put on the boiler for HW at least an hour before you get up? If your boiler is not very powerful it may have difficulty heating the house and the cylinder at the same time. This is usually mitigated by heating the cylinder first.
If you feel the horizontal pipe coming out of the top of the cylinder, it should always be hot, unless you have used up all the hot water. If you ever find it to be surprisingly cold, it suggests you have a faulty mixer tap allowing cold water to run into the cylinder. This will also cause the loft tank to overflow down a pipe that probably comes out under the eaves.
You can improve the insulation of the pipes and cylinder a bit, but that is not an urgent job and will not be your main problem.