Attwood's performances in Formula Two prompted Alfred Owen, the proprietor of BRM, to offer him an opportunity in his works Formula One team. His first outing for the team was in the non-Championship News of the World Trophy race, at Goodwood, in which he took the BRM P57 to fourth place, the first non-Lotus finisher and the only car to end on the same lap as Colin Chapman's fleet winners. Attwood's second Formula One outing was in the 1964 British Grand Prix, driving BRM's experimental four wheel drive P67 model. Having been the project's test driver Attwood did manage to qualify the overweight car, albeit in last place on the grid. However, as the car was principally intended as a rolling test bed, BRM decided to withdraw the P67 prior to the race itself.