Essentially, the P.1127 was designed around the Pegasus. The available thrust was utterly marginal to demonstrate VTOL and transition, so the P.1127 was the smallest and lightest airframe that could be wrapped around the engine and maintainability be damned.
The Pegasus itself was designed to demonstrate the concept of a bypass engine with a large forward fan fitted with rotating nozzles, enabling VTOL. The first Pegasus sketches did not have vectoring on the core exhaust. I recall reading somewhere that Hawker observed to Bristol that the Sea Hawk had a bifurcated exhaust, so why waste that vertical thrust? The original idea was to match an Orpheus with Olympus compressor stages, but there can't have been much left of either original by the time the BE.53 ran.
Where it gets interesting is that Bristol was introduced to the concept of vectored thrust via France's
Michel Wibault and his Gyroptere. This had four shaft-driven lift-cruise blowers, and the most powerful European turboprop at the time was the Bristol Orion. Stanley Hooker didn't like all the shafts and gearing and saw that a direct-drive axial fan with moving nozzles would be simpler and would also supercharge the core engine.
And, many years later, Paul Bevilaqua looked at the in-line tandem fan STOVL concept, which presented some nasty flow-switching issues, and decided to turn the front fan through 90 degrees with a clutch and gearbox, and separate the airflows, and so the F-35 was born.
And now you know the rest of the story...