I think £120 is good value for the alterations you're seeking. I would feel lucky to get my alterations done for that.
This. Most shops do an inclusive package which usually starts from £100. My BF needed just an inch of organza taken up off the bottom and they tried to charge her for the standard £100 package, she kicked up a stink but even then only got it down to £80.
I think alterations via shops will only get more expensive as shops are losing out to pre-owned dresses now, and also people are catching on to the fact that once the dress is paid for, the bride is actually free to take it away and organise her own alterations, which isn't something shops like to let on about.
All in all, suck it up or find someone private but it will probably turn out a very similar price - plus if anything went wrong (like me and my train getting cut off, ha!) you'd have more recourse via a shop.