(1) [to a minor] Will your parents let you go swimming (with me) now?; (2) [to an adult] Do you have time to go swimming (with me) now?

А вообще возможно (не сейчас)?
ходить плавать / на плавание

I’m having trouble imagining a context for that. Let’s say you’re on a long cross-country drive and you decide to stop for the night. The hotel has a pool, but it seems to be closed off and there are no people in or around it. We would never ask “Can you go swimming?” in this context. We might ask “Is it possible to go swimming?”, but would sooner ask “Is it ok to go swimming/use the pool?” Is that what you meant?

"Can you go swimming in the sea during the UK lockdown? "

" Can you go swimming with a tampon?/ on your period" (sorry)

"Can you go swimming or use a pool or hot tub during the Coronavirus epidemic? "

"Can you go swimming in Greece during October? "

Ah, ok.

But in all those cases, they express the circumstances. Here it’s just “can you go swimming” which sounds like a little boy taking to his friend.

Согласен, но ты знаешь как редко здесь можно увидеть полное предложение.

Еще пример
You need more exercising. Can you go swimming (for instance)?
??

That wouldn’t sound too natural because GO -ING usually refers to leisure activities/individual outings (albeit repeated). Here we would most likely say “You need more exercise. How about swimming? Would you consider taking up swimming? Is swimming an option? etc.

OK, thanks.

👍🏼