Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Newcastle upon Tyne to Belize City as you normally would.
Newcastle upon Tyne (NCL) is the airport you will fly out of if you book a flight from Newcastle upon Tyne to Belize City. You should not have trouble finding flights to Belize City from Newcastle; the airport handles an average of 0 outbound flights to Belize City per day. You should also have plenty of airline options to choose from since 2 airlines have flights from the Newcastle upon Tyne airport to Belize City. It is only a 5.4 mi journey to Newcastle (NCL) from Newcastle upon Tyne’s city center.
You don't need a visa for Belize if you hold a multiple-entry visa or valid residence permit for the United States or Canada, or hold a valid multiple-entry Schengen visa.