Seeing as that valve on the towel rail is/has been dripping, I'd want to drain the system enough to get the top floor empty and do all the work you need on that floor at once. It would be easy to connect to the filling loop valve on the right in your picture with a 1/2" female hose adapter for partial draining. I'll have a look in a minute to see if I can find a source or at least a picture of one for you.
Here is a list of the steps you need to take, quite a few but take your time, step by step.
1. Attach hose to filling loop valve on the right and discharge end in drain/sink/outside any safe place.
2. Open the valve connected to hose and drop pressure to zero.
3. Go to top floor and open bleed vents allowing air into system at top and it will start draining. Don't worry about the jammed valve, it'll drain through the TRV side.
4. It'll take a bit of guesswork but drain enough water so top floor rads and pipework is empty and close the valve you are draining through. You don't need to completely empty the system.
5. Close bleed vents.
Now you can work on the top floor without fear of water escape. Change the valve on the radiator that got chewed up. Replace the towel rail valve with a lockshield, like for like. Keep the new valves also closed for now. It's a bit hard for me to tell the sizes from the photo. Is that a 10mm pipe feeding the towel rail?
Now would be a good time to add one bottle of corrosion inhibitor through the cap on the top of the towel rail and then replace cap. Keep the inhibitor trapped in the towel rail, valves closed, incase you have a leak on a valve connection. You don't want to have to drain down again and lose the inhibitor.
6. Remove garden hose and reattach braided hose to filling loop
7. Refill system to 1.5 bar
8. Check for leaks at valve/pipework joint. If happy open the radiator valves and bleed air. The pressure in the system will keep dropping as you remove air so you'll need to top it up to 1 bar and keep bleeding rads until water issues from bleed screw.
9. Have a final look at all your radiator connections. If everything is still sealed properly, use the filling loop to fill to 1 bar with the system cold after you've finished bleeding rads.
10. Turn on your heating and check rads get warm.
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This is the type of hose connector you need to connect to the valve for draining -
1/2" BSP Threaded To Barbed Brass Fitting