Both TWA and Pan Am used their 747-100s on this route in the 1970s, but BA never put the 747-100 on LAX, it was considered beyond range. It was a significant part of why BA used to do a daily hire of an Air New Zealand DC-10-30 for several years in the late 1970s on the LAX route until their Rolls-powered 747-200B came along (when this was the first route they were deployed on).
There were apparently a range of concerns by ATC at Heathrow on warm summer afternoons over the prospects of an engine failure on departure, by no means unknown on the early JT9D 747s, especially if they were on easterlies heading out over the urban area, over the performance of these US operators' departures, inevitably at MTOW. Once they had gone everyone breathed a little easier, and there would be comments about departures "via the Piccadilly Line", or when Pan Am were still doing the route (they gave it up around 1975 in a route swap with TWA) that it was a "Hedge Clipper".
Those Air New Zealand DC-10-30s could still be somewhat marginal themselves, and needing to do a fuel stop, typically at Prestwick, was not unknown. The normal first course of action was to leave off some freight.