what about this scenario ?
BIS got chance to experiment with Solid and liquid fuel rocket engine thanks do loophole in Law, with help of university of Oxford.
in 1930s they build and tested several designs successful, what rase interest of Military for possible use against Germany.
1939 Hitler invade Poland, Britain declare War to the Third Reich, BIS and Oxford team take over by Military with higher Budget.
In the Blitz face Luftwaffe, RAF with fighters equip with rockets, while German bomber are attack with ground air rockets
The Third Reich counter with V1 and V2, Britain answer with there Avenger rockets
1945 The Nazi capitulate, found V2 are brought to Britain analysed by BIS.
Sadly between 1945 to 1951 stagnate the development, do Labour Government
1951 return of Churchill as Prime Minister, one of his first orders are program for MRBM against USSR.
Again work BIS and Oxford for Military, on Black Streak program for solid fuel MRBM
To test this MRBM, BIS proposed to launch "Satellites" into orbit, to obscure true purpose of Rocket,
1955 Britain start unintentional the Space Race, join by USA and USSR.
Now British politic are eager to continue the success since Britain was first in Space
and find partners with Australia and Canada for join venture Space program
while Prime Minister Anthony Eden under influence of amphetamines, declares:
We [British] are conqueror ,we climb the Mount Everest first, we fly first jet plane, we launch the first Satellite,
We shall do thing never see before, we shall land on Moon first and put union jack on it !
So started the Moon Race...