Not necessarily. The SR-71's range was 3250 miles, which is ~10% less than the NYC to LHR distance. It might encroach bingo fuel, but maybe not. If they refilled to completely full in New York, they could either refill or land once reaching the London area.
The link posted by @RcouF1uZ4gsC contains a graphic that shows a 'refuel' point over the mid-Atlantic and the linked article says this:
> ...it needed a minimum of one in-flight refueling, thanks to the astonishing amount of fuel it consumed while in afterburner. ... Having already surged across the Atlantic in a sustained supersonic cruise at speeds in excess of Mach 2.5-2.6 ... Sullivan began a descent that would bring him to the only refueling point of their mission. After tanking up, the duo climbed away and once again nudged the throttles ahead. The time taken up by the in-flight refueling was deemed virtually negligible when the Blackbird flew through the London gates.
So, according to the linked article from @RcouF1uZ4gsC, yes, the record time included one refuel during the NY to London route.
Most long distance Blackbird flights involved taking off at low fuel weights, then refueling shortly after take off. So yes, the record probably involved refueling over the Atlantic.
Then, if they got a full tank of JP-7 around NYC, that gives them a one-way range of 3250 miles, which is less than the 2999 nautical miles of LGA -> LHR. The SR-71's range is a hair over 3200 nautical miles. Getting a full tank of the special JP-7 or landing around LHR would be do or die time, because that would be just about bingo fuel.
