Look into covid tests for travel, they normally offer overnight results. Fit to fly can be done over the phone with an emailed certificate within a couple of hours.
On what basis do you live in Europe? The logistics may well change from January 1 when we finally split from the EU. If you're a legal resident, you should be able to apply from there - otherwise i fear you might need to return.
If you're not sure, put a reasonable guess and attempt to stick to it. I think I'm right that, if your insurance runs out after less than a year (for example 49 weeks), you'll get stamped in until it expires, ie not the full year.