AHC: Give the UK a space program that's up there (in the literal and metaphorical sense) with NASA and the Soviets, with a PoD as late as possible.
Having a British space program that plays in the same sandbox as the United States and Soviets in a world that broadly tracks OTL is an ASB proposition. There are ways to get there, if you really want it, but they involve things that have the practical effect of butterflying away both World Wars as we know them. Britain's certainly capable of having the equivalent of the Ariane I flying years before the founding of ELDO and once you've got the capability to throw a few tons of payload into LEO, the question then becomes whether you want a manned spaceflight program rather than whether it's possible to have one. (And we probably want one, because everything's better with a British accent.)
But that requires getting Blue Streak off the ground and that does face real hurdles. There is the most obvious one in the Treasury, which is the common enemy of all British aerospace projects in the Fifties and Sixties. But Blue Streak is itself a money-pit beyond
just the half-billion pounds necessary to get the thing flying and acquire enough of them to satisfy the deterrent role, between the need for hardened silos and the fact that a storable-fuel "upgrade" program -- that is more likely than not an entirely new rocket engine development program -- will be required even before the first operational missile is delivered. And to confound this even more, there's also a ticking clock, as Blue Streak needs to be mature and operational by the time Polaris starts demonstrating its progress, because once that happens the Royal Navy will be able to make the same arguments it did OTL and, if not already bought-and-paid for, the Treasury will want to wash its hands of Blue Streak. Which would be a problem because Polaris, despite being a great IRBM, is not a great building block for a launcher. At least not without a significant redesign, which runs into the buzzsaw of the Treasury once more.
My personal pet solution to the problem is to have the USAF pay for it, with Blue Streak filling the role of OTL's Thor. It's not as far-fetched as it might seem, as at the time Thor was given the developmental greenlight, there was a prior agreement between the U.S. and U.K. regarding the special relationship's division of labor for ballistic missiles, with the former to focus on ICBM development while the latter would focus on IRBM development, with the U.S. chipping in 10-15% of what would become Blue Streak's budget. And I believe there was at least some high-level discussion of pursuing Blue Streak in response to the recommendations of the Killian Commission, but it was abandoned in short order due to Not Made Here-ism and a lack of confidence in the British ability to effectively manage the project.
At any rate, once you have Blue Streak, you've got a decent foundation to build-on as a booster, given its dimensional and thrust similarities with the early Titan first-stages while being powered by essentially the same engine as the early Atlas's half-stage. Give it a decent second-stage -- license-built Centaur is never a bad choice -- and you've got yourself an alternative Black Prince that's capable of lasting for decades with incremental improvement. And if you make things a properly Commonwealth exercise, you can even net yourself James Chamberlin to design you a Fancily Accented Gemini capsule in the process. (Or just straight-up license Gemini too, as it's realistically not going to be until the early-to-mid-Seventies before the money's going to be there for even a modest manned program, even if you're licensing everything to minimize developmental costs.)
I think their propellant choice was fine: non-toxic, non-cryogenic, high density and very reliable.
So why has everyone else ignored HOOH for rocketry? Blue Origin even gave up on that fuel, a decade ago.
H2O2 does have a good deal to recommend it in comparison to other hypergolic fuels. But the stuff is only really useful in distressingly high concentrations and, while it can be well-mannered under very specific conditions, you can never really trust it to behave. Because the stuff's an effective monopropellant in its own right, decomposes into stuff that it readily explodes with, and is prone to runaway decomposition if not treated with the utmost respect. If somebody ever suggests you procure 5,000 gallons of 90% concentration H2O2 for the purposes of dropping a rat into "to see what happens", you ought to run the opposite direction as fast as possible. (This is, apparently, an idea that somebody at the Pentagon floated to the Naval Air Rocket Laboratory in the early Fifties.)
More practically, H2O2 has floundered because it's a fairly low-performing fuel. Keroxide engines have got an optimal vacuum specific impulse of 320-330s -- which comes from Astronautix, so should be taken with a grain of salt, but is almost certainly close enough for our purposes -- which is inferior to that of kerolox's ~350s. And once the usual engineering and application losses start to bite, the performance drop really stings. Black Arrow's pump-fed Gamma 8 had a sea-level ISP of 228s or thereabouts, which is playing in the same sandbox as pressure-fed kerolox engines, with an oxidizer that is far more temperamental than liquid oxygen. While for upper-stage applications, the 260-270s from the Gamma 2 is having to contend with the elephant in the room that are hydrolox systems. As specific impulse matters even more for upper-stages and the gains from hydrolox are such that it's still a markedly superior choice even with all of LH2's handling problems.
How do a couple of stunts using captured V-2s translate into a meaningful British space program? The fundamental problem with a British space program has been a lack of funding and, while notching a few aerospace firsts might garner a bit of political support, the British electorate is not going to tolerate for long a government that chooses to pump hundreds of millions of pounds into the aerospace sector to launch a space program at a time when rationing is still on and those are scarce pounds that could be expended building-out the Post-War Consensus's welfare state. You can kinda-sorta elide your way past that with Blue Streak because Britain remains a great power and, by the Government's own logic, it needs ballistic missiles with which to deliver its nuclear weapons, even if those are pounds that the Government would prefer to spend on other things. But for much beyond that requires the rolling a lot of 6s/10s/20s (depending upon what dice system you think history's using) and even then you're still not going to accomplish a ton.
The usual alternative to H2O2/kerosene is hypergolics, which have been dubbed "explosive cancer" for a reason.
I believe "explosive cancer" specifically refers to UDMH/N2O4, with the dinitrogen tetroxide in particular providing the "cancer" part of the equation. The various flavors of hydrazine and nitric acid tend to just explode or melt your face, killing you horribly long before you could develop cancer. While the more exotic hypergolics -- like any of the ones involving a fluorine-based oxidizer -- tend to kill you even more horribly even faster than hydrazine/nitric acid.