Thanks.
I thought I wrote a reply earlier but it doesn't show.
When did the heating and HW last work properly?
Your system looks to have been good quality and well installed when it was new, but looks neglected and needs repairs at least, and preferably updating. Without wishing to be impertinent, is cost a problem?
It seems to me that some (or all) of your problems are caused by the old and faulty radiator valves, and that this should be the first thing to rectify.
It doesn't need a gas installer, so ask around for personal recommendations of an experienced local plumber. I'd replace all the old ones. This will involve draining down the system for a day or so, which at this time of year should not be too bad, as I see you have an immersion heater on your hot water cylinder. It would be preferable to put a thermostatic radiator valve on one end of each radiator, since you have no wall stat and presumably no motorised valve.
If you are willing and able to get into the loft a week or two before the drain, and add a cleaning chemical to your feed and expansion tank, after ladelling out all the mud and sediment and wiping it clean, this will help the drain to wash away more of the old dirt from the system. It's a cheap and easy move wich I can explan more fully if you wish.
Have a look at your gas meter and see how many cubic metres of gas you are using per day. For Hot Water only, I would expect it, with your system, to be in the region a 1 per day. With heating it would be more.
After fixing the radiator valves and getting the system going, you can look into updating the controls and insulation on the cylinder and the pipes, especially the ones between the boiler and the cylinder, which will improve efficiency and cut costs.
I don't see a timer. How do you turn your boiler on and off?