Hi there,
I've been in your shoes. Your choice of licence should be dictated by where you can work. The EU/UK thing is of course a clusterF so not sure what to advise you there.
I can say that you will absolutely not get a green card or work permit in the US as a pilot. Unless you have any other plans to get US residency (marriage etc.) You have absolutely zero chance of landing an aviation job in the US.
the one exception is to do a zero to CFI course at very few pt141 schools who will issue a student/work visa. This will allow you to instruct for 12months, but after that it's back to your home country and you will not be able to find a company to sponsor your residency in the US. They legally can't and wouldn't waste their time to sponsor you especially with so many furloghed pilots. Then you will be back at your starting point and unable to get a job with out converting. Check any job requirements for any US job, they all say "Right to live and Work in Us"
on the other hand both EASA and FAA are equally recognised in Asia, middle East and Africa. I would say FAA is cheaper and easier to maintain than EASA.
Finally my best advice would be to hold off on any training and investment for at least 12 months due to the situation. Even if you had a work permit for any of these places, there will be no jobs for a long time and plenty of jet experienced pilots in front of You.