The change was made in 1965. The last ABH on the first Monday in August in England and Wales was Monday 3 August 1964. dzug is quite correct in that the change was only formalized in 1971 with a change to the law.
From 1965 to 1970 the holiday fell on the last Monday in August (where it remains) except for 1968 and 1969 when it fell on 2 September and 1 September respectively.
The background to the change was twofold: the tourist industry had been pressing government to make the change almost since the end of WW2, in order to extend the "season" (remember that hardly anyone went abroad in those days) and secondly, many factories and other industries had a "shut-down fortnight" which was often the last week of July and the first week in August and, together with the Bank Holiday, this put intolerable pressure on resorts, roads, and railways for two weeks, so the government moved the Bank Holiday in order to spread the load a little.
Whit Monday moved with Easter (Whitsun is seven weeks after Easter) and this meant that Whit Monday could fall any time between 12 May and 14 June. Since the summer half-term for schools invariably fell in the holiday week, this caused problems for schools, especially since by the 1960s many more pupils were taking public examinations. For this and several other reasons, the holiday (but not the religious festival) was "fixed" to the last Monday in May from 1967 onwards (and like the August holiday, the change was made in law in 1971).