Now they're in a refugee hostel in Bristol or Plymouth or wherever. What are the UK's responsibilities here?
The same as any other asylum seeker in the country. Make sure they are safe, assist them with their application to go to USA, and if USA reject their application, it's the UK's problem. We can either deport them back to the country they arrived from, or take them in. Depends on their circumstances and the law.

I think we can agree that giving them food and shelter is on us while they're here, so no arguments about that.
Indeed, and we do provide food and shelter for refugees and migrants. Not a problem.

But, do you also think the UK is responsible for determining whether this refugee's application for asylum will be approved by the USA before allowing them to leave the UK for the USA?
Yes. We should not send them to USA without the USA approving of it.

who are we to decide how they choose to get there?
You understand we are obligated to feed and shelter them, yet to fail to recognise our obligation to ensure their safety. We should most certainly stop them from getting in a kayak and leaving the Cornish coast for the Atlantic. Do you think we should do nothing about that if it were happening?

And if so, how do we enforce it?
We patrol the waters, assuming there is a problem with people trying to leave the country like this. We have to do something, right? We can't do nothing. That is highly irresponsible.