Can I legally own and fly fighter jets in the UK?
I don't know of any privately owned Hawks, but there are many ex-RAF jets in private ownership, like the Jet Provost, Lightning, Gnat, Meteor, Vampire, and others I can't remember. The Ministry of Defence sells fixed-wing airplanes and helicopters occasionally at auction; when they retire the Hawk T1 you could be able to pick one up.
This CAA document goes over many of the particulars, a quick summary is that in almost all cases they would be under a permit to fly. This would require an inspection to clear it for the permit, and an (un)healthy dose of paperwork. I know of no way to get these aircraft certified, but you can get them cleared for night and IFR flight with extra paperwork and continued inspections, provided the aircraft is equipped for it. You can also get a permit to fly faster than 250kt below 10,000ft, which in a jet like the Hawk would be an advantage. Flying abroad on a UK permit aircraft is just more paperwork: you have to get permission from all the countries you'd fly over and land at, which is generally a formality, and restricted to day VFR.
One of the challenges you would find is keeping the plane airworthy. There are a few organizations that restore and maintain ex-military jets; you'd want to find out if they'd be able and willing to take it on for you before you buy it.
Of course you'd also need an awful lot of money. Jet Provosts (50s era trainers) are at the low end of jets as costs go and they still cost £600 per hour to operate!