Is the £350 just for your appointment and prescription, or does it include the methadone as well?

Not that I really know the going rate for these things but if it's just the appointment/prescription that does seem rather high. But if your medicine is included too, well, that costs.

As others have said, free medical help has nothing to do with your nationality. It depends on your being resident in the UK (you don't even have to be a tax payer, but you at least have to live here, which seems fair to me)

You are not a resident of the UK nor a member of their health system. You are seeking an emergency RX for a controlled substance. If you were in NYC I beleive you might well be charged a good deal more than that and have to go through a huge number of hoops to obtain it at all.

You family troubles are unfortunate - but no one took advantage of you - they provided a service that solved yuo problem.

Methadone is a very cheap drug...pennies per milligram, I believe.