You should be able to get a flight from UK to Cancun Airport.
The easy way to Belize is to then change to the Tropic Air flight to Belize City (Belizean airline -- google it for schedules & fares).
Another easy way is to arrange for one of the transfer services to meet you at the airport and drive you in comfort to Belize. Expect a road trip of several hours. Or you could go to Chetumal with an ADO (Mexican bus company) express bus and have the transfer service pick you up there.
The cheaper way is to head to Belize by bus or water taxi.
Get the ADO overnight express bus from Cancun City bus terminal to Belize City.
Go with an ADO express bus to Chetumal, and then continue to Belize by either a Belizean bus (old school bus) or by water taxi to Ambergris Caye. You might have to spend a night in Chetumal.
Have a look at the BelizeBus site for more details...http://belizebus.wordpress.com/