What type of tree is this?

Hi all!
Sorry it's been an age. We've moved house... to a MUCH bigger garden. Pics to follow in the near future, no doubt, although I doubt I'll really get to tackle the garden for another year, and it will be fun to watch what grows over the next 12 months before we make decisions.
Anyway, in the meantime, can anyone identify this tree?

It is, unfortunately, right by the neighbour's wall, and she wants it removed. It's blocking light from her window, and (she believes) causing a damp problem in her property (?!).
It is also too big for its current location, so realistically, it probably does need to go.
But I'd like to know what it is, before I make any irreversible decisions. Ideally, if we do chop it down, I'll try to take cuttings and relocate them somewhere more suitable.

 

It's Garrya elliptica, a lovely shrub often grown against walls

 

Lovely, thank you. Sadly, I don't think it can stay where it is, although I agree, it is a lovely shrub.
Any ideas how to take cuttings?

 

Your timing for cuttings is just about right, now is the time take semi-ripe cuttings before the new growth fully hardens off

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