I am in London. Nice riding up there, but a bit far for the weekend!
I have built up a full set of tools over a good many years, and whilst some of them (headset press for example) don't get used that often, I have found that I can get by with the following and use a shop for the bits that require expensive and rarely used tools:
Frame
Shop - Fit Headset, Chase and Face bb, disc mounts and rear mech threads and the seat tube if they have the tool. Sometimes paint gets caught down there and it can score your seatpost.
You - Nothing
Forks
Shop - Fit Crown Race
You - Trim steerer once you have decided the height (ideally purchase a saw guide, but a plumbing pipe cutter and metal file will do it well), fit star-fangled nut (ideally with a setting tool). You might want a shop to do this for you. Adjust the tension and the stem with a 4mm and 5mm allen key.
Controls (seatpost, saddle, stem, bars)
Shop - Nothing
You - Should be able to do everything with a 4 and 5mm allen key. Don't forget to grease where you need to
Wheels
Shop - Nothing
You - Possibly fit rotors (cassette tool for centre lock, bb tool for centrelock 20mm axel, Torx T25 tool for 6 bolt), fit tyres and tubes (a good strong pair of thumbs should suffice!), fit cassette (cassette lockring tool)
Drivechain
Shop - Nothing
You - Fit bb (HT2 spanner, and lots of moly paste), fit mechs (5mm allen key), fit chainset (possibly HT2 bearing tensioner, 5mm allen key if Shimano, 10mm if Race Face), fit pedals (usualy an 8mm allen key nowadays), fit shifters and brakes (4mm and 5mm allen key), fit and tension cables (cable cutters, 5mm allen key), fit chain (chain splitter, but buy a chain with a quick link, and for breaking the chain once a mini tool chain splitter should be enough).
And that's about it. Other than a decent set of ball ended allen keys (Bondhus are highly recommended and will last years), there are very few specific tools you'll need. I would recommend buying the bb tools as unfortunately you might need those every now and again! Also, the cassette lockring tool, particularly if you use centrelock rotors. You'll also need a chain whip to remove a cassette but that can wait a while yet!
As satisfying as it is to do everything yourself, the cost of the headset tools and prohibitive and you are probably best to leave that to a shop, but everything else is simple with an outlay of probably £50-£60 on tools that you will use over again. Cheaper than the labour for a shop to build it up.
The only other thing might be a brake bleed kit if you decide to shorten your hoses, or need them longer if your old ones don't fit. They aren't expensive, and something you should get to grips with anyway for ongoing maintenance.