Given the 707's range was about 3500 - 3750 miles and the distance from LHR to LAX is 5400 miles, there would have had to have been a re-fuelling stop.
Sorry, incorrect information about the range/distance.
We speak of distance in air transportation in nautical miles (not statute) so the distance from LHR to LAX is more like 4800NM.
In addition, as I have flown (as a Captain) some of these older B707 intercontinential models, LHR-LAX was at the extreme end of the distance possible, however it was done on a regular basis, providing however...optimum levels were possible from ATC, a min distance routing was selected, the weather at destination was forecast to be favorable, re-dispatch procedures were used (to minimise contingency fuel uplifted) and finally, the required distance to the alternate selected was minimal.
The LHR-LAX sector normally would fit this profile above, so yes, the route was normally flown non-stop, on some days.