My old cherry tree looks like its end is coming near. Last year it
produced quite a bit of fruit but now it looks dead and all branches
seem dry (as if dormant) except for 2 or 3 small branches high up on
the trunk which managed to develop leaves and a few buds. This tree is
a one-of-a-kind and is the only one left in rhe whole neighbourhood
(there were many before, all in a row), and it would be a big
disappointment if it were to die without an attempt to graft/bud/etc.
to "clone" it.

Try the following nursery: http://www.bernwodeplants.co.uk/ They
specialise in preserving and propagating traditional fruit tree
varieties. I believe they offer a grafting service for people who want
to clone an existing fruit tree.