Hi! At the moment, to be honest, I don't know. Well I do know of one place but I wouldn't recommend it! I've got an Amazon voucher to get my hair cut at a fancy salon in the city centre and I intend to go soon, but I won't be talking to them about being trans.
To be honest though, I doubt youd have much trouble finding a place in Manchester. With some of them you can email beforehand, so you could always try asking that way? There is a hairdressers in the gay village called, er, Village hair

you could give them a try?