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My ancestor died in the Oxfordshire County Asylum in 1903. The county archives were happy to scan all of his records and email them to me because (a) his death was over 100 years ago, and (b) no other patients were mentioned. They were charging £1 per page a few years ago and it was well worth it.
From what I've read of other people's experiences, I would say that's fairly typical around the country. It won't do any harm to ask

Carol