BOAC got into Transatlantic charters (and to a lesser extent to the Far East) when the 747s came along and they were left with surplus 707 capacity, which ws reconfigured to Economy only. Pan Am and TWA were the same (I recall TWA 707s coming through Manchester, never a normal scheduled point for them). There was a specific subsidiary, BOAC Charter Ltd, set up to do the marketing of these, but the ops side was handled by the regular organisation.
Favourite points for BOAC were New York, Toronto, Los Angeles and Oakland (for some reason charters to the San Francisco area always went to Oakland). This traffic disappeared again as more 747s came along and they filled up the back with cheaper Economy fares. Some of the BOAC Economy-only 707s also got used at this time on some schedules, Toronto being a favourite for this.
It was difficult to keep to regular back-to-back flights in this market so there was a fair amount of unproductive positioning required to handle these flights, as described above.
Regarding the short-haul operations, in the early 1970s BEA, and then the Euro Division of BA, had a couple of shortfalls of capacity, and, besides using up whatever BEA Airtours had spare in the wintertime, which kept the Comet 4Bs on schedules from Heathrow well after you would think they had ended, they used both 707s and VC10s from BOAC (and later BA Overseas Division) on Euro routes. There were VC10s from Heathrow to places like Amsterdam and Rome as substitutes for sufficiently long that they appeared as such in some timetables.