Rosie, that's a terrible situation to be in. Because it's not listed, I'm afraid I can't think of any other avenues. Perhaps someone else can. I do hope so.
What I can do , if it's any help, is to point out a few practicalities that might bring the cost down.
First, I would suggest a small builder, one who is not VAT registered. The job is largely labour, so the VAT addition is substantial.
I'm guessing at the type of wall, but mostly they're built in a way that does not need a "Stonemason" of the monumental variety. These days, trades are taught in brick/block laying only. They're not what you need either, unless they have had some experience - many have.
See if you can find what used to be called simply a "Mason". I guess they'll all be in their 60s or 70s now, but they're still around. Mason's apprenticeships included brick, block, and stone.
One of the old-school Masons, could happily plod along and do this on his own, even I could, and I know of one or two who do this kind of thing easily.
I'd like to know how much square metreage you have, as I could do a hell of a lot of walling for £10k.
One other thing. It's not listed, so no restrictions on how it's done. I guess you'd like to keep the face showing in the alley, but what about your side? I'm sure you'd prefer it to be stone facework, but rendered blockwork can easily be made to blend in with the old in the right hands. It's not uncommon.
Much reduced labour cost. The wall would only need to be faced up on one side - plus some sort of capping.
None of this may be relevant, but good luck anyway.