500mm x 500mm x 500mm is big enough for a feed and expansion tank but nowhere near big enough for a cold water tank (cistern) that is supposed to run a bath.
Water tanks are generally plastic now, and should have close-fitting lids to keep dirt and insects out, and the overflow is also screened now.
I think yours are overdue for a replacement. It's a fairly easy plumbers job, or an enthusiastic DIY handywoman like yourself. You don't need to pay gas engineer rates.
The new ones are likely to be quieter filling, as the fill valves will be new and probably a modern design.
The plastic tanks are flexible enough to squeeze into a loft, it might be that the previous person fitted little ones he could get through the loft hatch. A long, narrow one is called a "coffin tank." Screw down new pieces of 18mm ply across the joists so the tank is standing on a flat surface, or the plastic tank may deform or split under the weight of water.
This would be a convenient time to give the CH system a good clean and flush, because you can drain it down while changing the F&E tank, and will have all your plumbing tools to hand. Sentinel X400 is quite good at loosening the sludge so it washes out when you drain and flush. Normally you woulde circulate it through the radiators hot, but you can turn down the boiler and do it cold unless we get a spell of hot weather. You can circulate it for up to 4 weeks before it starts to lose its detergent quality and the dirt starts to settle again. have a system filter such as a Magnaclean fitted before you finish cleaning. Ask the plumber to show you how to empty the dirt out, it is no harder than emptying a hoover bag when you know how. At first it will need emptying weekly or monthly, but because it captures all the circulating particles, the system willl gradually become cleaner and once a year will do.
On final clean fill, add X100 which will reduce risk of future corrosion (it is not a cleaner) and X200 which will slowly dissolve the limescale deposits. It does not have to be drained out. You'll need one litre of each chemical. Around £16 each.
Here are some examples of tanks. I think 125litres is the smallest sensible size for a house with bath and shower. Mine is 220 litres (50 gallon).
Your Profile 50 is about 15kW, which is sufficient for an average house and cylinder with reasonable insulation.
You can measure your radiatrors and work out their output, but it might be in the region of 1200W each. A 1500W radiator very rarely runs at max power, except if you have been away in winter and the house fabric has got cold. Your radiators should have TRVs, of course, and should be balanced so that they all heat up at about the same speed (except the one in the room with the wall stat, which should be adjusted to heat up slowest)
Check that the electric immersion heater in your hot water cylinder is in good working order, and does not overheat the water, before starting the other plumbing work. In summer, an immersion heater is all you need, though in your case it will give a shallow bath. Leave it on all day and check that the hot tap is not scalding. This is very dangerous but easily repaired. Do not leave it on at night or unattended until you have checked this.