I doubt it. IIRC much more damage was done by German bombers, that didn't seem to dent support for Bomber Command during the war or afterwards. Now I suppose you could make the argument that bombers are an established capability versus a new one, but for the scientists and military men who would be making the decisions I think they'd be fairly hard-headed.
What if, via the west being more open with the Soviet Union, post war Russia is no longer seen as big of a threat as it was by Truman? It would likely remove the pressure for nuclear capabilities to match that of America and the USSR. And in terms of aircraft development, the money was definitely there. The Bristol Brabazon for example, set back the taxpayer some £12,000,000 in 1949, which would be worth approximately some 542 million pounds.
Further the X-15 programme, costed approximately 300 million dollars in 1969 currency. In 1949 that would have been in the vicinity of 150 million pounds, or something like it.
I think funds could have been spared for a manned programme, however development would likely have been much slower than those of NASA and the Soviets, especially when Britain would have to put prioritise the rebuilding of the country, however the space programme would likely improve morale an awful lot. I would go with NASA's estimates for a manned Megaroc Launch by 1952, given that the budget would likely be described as a "trickle".
Likely the cause for launch delays, in addition to the highly constrained budget would have been problems with the engine. Running under prolonged burn times, and taking into account the general unreliability of the A4 engine, would likely have caused the Megaroc programme to stall, as solutions would be found.
I still think it would be possible, if the scenario it is based on came to be.