that boiler doesn't need topping up, it is fed from a small tank in the loft.
if the programmer has a digital display it must be quite modern. Possibly some incompetent oaf has fitted one intended for a combi boiler, where there is no time setting for the HW. However if we knew the make and model it is possible that there is some way of controlling it.
As you have a tapstat, your system has (or had, if it was later changed) pumped flow to the radiators, and gravity flow to the cylinder. Gravity flow is slow so may take a couple of hours to warm up the cylinder. It will be more economical if you insulate all the hot pipes between the boiler and the cylinder with Climaflex or similar, otherwise they will act as long thin radiators all summer.
Your system ought to have a room stat which turns the CH pump on and off according to demand for heat. It sounds rather as if there is no stat so the pump will run all the time the boiler is on, even if the house is warm, which is bad. Even worse, it sounds as if there is no separate control of the HW and CH, so that to heat the cylinder, you turn on the boiler and the CH pump starts, even in midsummer.
Compounding this poor installation, your TRVs, you say, do not seem to work, so whenever you try to heat the cylinder, the radiators heat up. This will waste a lot of energy and heat.
I suspect that the person who fitted the (new seeming) programmer did not know what he was doing, so cannot have been competent.
The simplest and cheapest correction would be to fit a room stat so that it turns the pump on and off. Any competent heating engineer and most plumbers could do that in less than an hour. You can buy a simple room stat for less than £10 though better programmable stats are more expensive.