The most famous of place of public execution in Britain, Tyburn (modern day Marble Arch), saw it's last visitors hanged on the 29th August 1783 - William Ryland for forgery and John Austin for robbery.
However, public executions still continued (only moved from Tyburn because the area was becoming more upmarket and local residents objected to the vast crowds gathered to attend) but took place at the Old Bailey, next to Newgate prison. The final public execution in Britain took place on the 26th May 1868 when Michael Barrett, an Irish Republican, was hanged for trying to blow up the Middlesex House of Detention but managed to demolish a terrace of houses, killing twelve and injuring forty.
After this, all executions took place within prison grounds.
Capital punishment, in regards to homicide, was abolished in 1965 - Peter Anthony Allen being the last man hanged for murder the previous year - and the final offences that still held that sentance were changed in 1998.