This could be a short story by Stephen Baxter and Simon Bradshaw entitled "First to the Moon". It is set in an alternate history where Edward VIII did not abdicate and Lord Halifax was still Prime Minister in 1950. The "Imperial Space Programme" made use of captured Soviet rocket scientists:
the detailed work and testing had been done by Sergei Korolev and his
handful of rocketry and aeronautical engineers, who had been snatched
out of imploding Russia just before German and Japanese troops had met
at the Urals in 1944.
In this alternate reality WWII indeed seems to have turned out differently. The Axis powers were victorious, the USA seems not to have been involved, and the only mention of Britain's role is a compromise with the Reich in 1942:
Even though the European war might have become a disastrous,
unwinnable conflict... there had been no particular honour in Prime
Minister Halifax's compromises with the Reich, made with the grinning
Germanophile Edward VIII at his side.
The moon mission, the HMS Captain Robert Scott, indeed runs into difficulties; the pneumatic landing gear is damaged and the rocket topples to one side shortly after landing. The story concludes with the surviving mission members looking up at the sky and seeing a craft launched from Earth - a nuclear powered rocket (the "von Braun"), launched by the Nazis, which they hope will rescue them.
It was first published in Spectrum SF 6 in 2001, and anthologized with other short stories by Baxter in "The Hunters of Pangaea". It is very hard to find a copy these days though. A version used to be available from a link from Bradshaw's website, but sadly it seems that link has succumbed to bit-rot, although it is still viewable via the Wayback Machine (kudos to FuzzyBoots
for suggesting this).