I do agree that they might balk at you asking for indefinite flexible leave following three weeks of flexible leave for your son.
(Again, not that it's my business, but are you planning to take him with you? Leave him in the UK? If the former, it might even extend the need for flexibility further.)
I think you need to separate out the desire to see your father one more time and the need to be there for probate. For the former, even assuming he does get COVID, does your father want to see you while he's in prison? For the latter, are you sure that you have to be there?
If you can solve one of the issues, you won't be dealing with indefinite. Let's imagine that you're right, and he's going to get ill. (It's not a given.) But if it's only about seeing him and someone else can get POA for probate, then you just need two weeks of leave; if you think he'd rather not see you, or if you can talk with him on the phone/some sort of video link, then it's just about going over for probate.
In either case, I think clarity with your work is what you have to do: here's the situation (my father has taken ill with COVID, although only once that's true), here's how long I'll be gone, here's how I'll make things up.
Alternately, and I hate to say this, but if you could live off family money, you might decide that seeing your father and being there is more important than your job. So you can go in and say, look, I need X months off to deal with family matters due to Covid. I'll take them unpaid, but I'd like to come back to my job if I can. If that means you have to replace me permanently, I understand.
But I don't think there are any jobs that would just offer you indefinite flexible paid leave.