Good, OTL situation in reverse - Europe rules, NASA merely follow.
Not only is an Anglo-French program on track to put humans on the Moon before the USA can, but Silver Star/Constellation clearly has the seeds of a reusable space launch system that makes it quite competitive with the proposed Pegasus. The schemes to recover and reuse Orion engines, or indeed to recover the booster stages in toto, are on the same track as I suggested could transform Pegasus.
The difference being of course that the recovered SS or C components would not be the stages that reach orbit, but rather the booster stages; to develop the ability to recover the engines required to reach orbit would be a completely new project, one that enables retrieval of components from full orbital speed, which would be much more challenging. But this is the challenge the Americans have taken on.
Meanwhile the Shuttle proposal that has been accepted in the USA apparently does nothing to recover the solid boosters, which correspond to the booster elements the British have attempted to make reusable. OTL the solid boosters of STS were recovered and refurbished--but this hardly proved to be much of an economic benefit, because a solid rocket is mostly made of the propellant.
And also because they would "splash" into the ocean and need to be fished out and returned to base--there to be broken into its segments which would be shipped over the continent to the Thiokol operation in the west, where they'd be filled with propellant (after inspection and refurbishment--the salt water float having done them no good), then shipped back full to Canaveral to be reassembled. (Clearly if NASA had been more serious about the solids being an economically reusable item, they'd have insisted on the facilities to refurbish and refill them being on site at Canaveral--and perhaps this could have enabled a less compromised structure for the boosters as well).
The proposals to recover Orion engines, or even entire Silver Star stages, on the other hand are considerably more advanced in this time line. A liquid fueled rocket typically will incur most of its cost in the engine; if a system can be developed to recover that engine and that engine can then be used many times, some serious economic progress is being made. If one can recover and reuse the tankage as well, then even if the upper stages are not being reused, the lower one will be, and the booster stage is the most massive by far--true of STS as much as of Saturn or Constellation. Mass roughly corresponds to cost.
Meanwhile, if there is a market for a reusable Anglo-French launcher system, funds should be forthcoming to not only perfect some version of reusable Silver Star, but to develop some degree of recoverability of the orbiting stage engine as well, or possibly the entire upper stage.
At that point, a launch system based on fully recoverable Silver Star (which implies at least partially recoverable Constellation) would be clearly superior to the American Shuttle, and to Pegasus as well--more recovery than the former, and the latter is not recoverable at all. Perhaps in response the Americans will develop liquid fueled boosters to replace the solids, and recover and reuse them, thus answering the implied challenge.
Honestly, I'd think that in such a TL, sooner or later the Americans are going to react to their slipping situation and do something dramatic to assert their superiority. Specifically, I can't see Nixon simply shrugging off the "Mouse on the Moon" scenario--note how vigorously the Kennedy and Johnson administrations tried to divert the British away from Black Anvil, and with it Selene. Failing to keep their lead by negative means, facing the fact that to discourage the Anglo-French project would be to rupture the most important alliances the US keeps, I'd think that economic woes or none, Nixon would kick NASA's Lunar program into overdrive rather than put it on the back burner, and leave the Shuttle as someone else's issue. Bearing in mind that he seeks to be re-elected in 1972 and not leave office until 1977, if the British and French can land someone on the Moon before November '76, the egg will not only be on his personal face but overshadow whichever Republican he has promoted as that year's candidate.
He would be strongly motivated to reverse this negative publicity with a prior US success, one which (assuming his re-election in 1972) would be clearly a Nixon and Republican legacy.
By this point, pretty much through 1970, the Americans would have to do something desperate to catch up and guarantee the ability to beat Selene in the event the European project makes optimal progress. Still, they have some three years; they are in a stronger position than the OTL US space program was in 1966 I would think. With the Saturn III in hand, brute force solutions that lack elegance or economy but can take advantage of the impressive mass to orbit should be considered; vice versa OTL NASA considered some real shoe-string ventures to barely place an astronaut on the moon and retrieve him; these ought to be dusted off and reconsidered in view of existing technology.
To accomplish a mission as elegant and capable as the OTL Apollo before Selene can reach the Moon would probably require panic budgets comparable to what NASA got OTL in the mid-60s, and that is probably not in the cards--though the President might make the case strongly enough to get it. But I would think that in an age where no human being has landed on the Moon yet, whereas the Soviets have done the stunt of a circumlunar flight, enough money to enable some sort of landing mission to be the first on the Moon would be forthcoming, and the program would take priority.
I can't see a President like Nixon or a country like my own in the early Seventies, simply relaxing and hoping it will be the British and French on a skimpy budget that beats the Russians to the first lunar landing, and resigning the whole race to those two and maybe the Chinese. While the race is yet to be won, I can't see Americans dropping out of it.
Though I suppose I could see them procrastinating along to 1969 or so before panicking and realizing they could lose, not just to the Soviets, but to their European sidekicks as well, and then realizing that the race has gotten real, that either competitor might pull it off before the USA is quite ready.
But not choosing, as Nixon has done here, to forget about the Moon Race and focus on a Shuttle program instead.