Probably one of the reasons they are quoting £370 is because they know exactly what to do and what to buy.
It depends entirely on the level you want, you will need a shifter, a rear mech, chainring, chain, cassette, inner, outter and maybe a wheel (maybe not depending on what you have already). Prices would vary from I think ~100 to up to what you have mentioned depending on group set level.
No one can give you a definitive list because it all depends on your preference and your current components. None of which anyone knows about.
TL;DR yes you can do it cheaper than that, but you need to know what you are doing and reading up on 1x set ups etc. will help you find out what you are doing.