Whether Apollo would survive or not is tricky, I would say NO. Apollo 1 and Apollo 13 only three years apart truly scared the shit out of everybody: from President Nixon to NASA Houston's Bob Gilruth, astronauts included (even the great John Young, damn it).
More generally, Apollo was not a practical / reliable / economical (think: DC-3 - 707 - 747) transportation system to the Moon. More like a pioneer vehicle: Bleriot XI or Spirit of St Louis.
The following is probably wishful thinking on my part and while some of the facts & figures may not be 100% correct, they are accurate enough for the purposes of this message, so here goes.
Without a Vietnam War there's no need to reduce spending on military spaceflight (unmanned as well as manned) in the second half of the 1960s, so my first piece of wishful thinking is that the Manned Orbital Laboratory project isn't first delayed and then cancelled outright. Therefore, 2 unmanned and a minimum of 5 manned MOLs are launched between the late 1960s and middle 1970s. I don't know what the effect on unmanned military spaceflight might be because I don't know enough about it.
The backlash against civilian spaceflight in general and manned spaceflight in particular still happens. However, my second piece of wishful thinking is that NASA will receive more money 1965-75 due to there being no Vietnam War.
Therefore, I think there will be a lot more money for unmanned space probes. The POD is probably too late to save the cancelled Ranger missions, but I do think we'll have at least one more Lunar Orbiter and several more Surveyors. There may be more Mariner probes to Mars and Venus in the 1960s. The follow-on Voyager Mars/Venus probe is still cancelled, but not until even more money is spent on it IOTL. The succeeding Viking project might not be put back from 1973 to 1975. More money may be spent on the original version of the Grand Tour, but it's probably still abandoned in the early 1970s in favour of the OTL Project Voyager.
More Saturn launchers, Apollo CSMs and Apollo LMs are built before their production lines are closed. At the very least SA-213, SA-214, SA516, SA-517, CSM-115, CSM-115A, LM-13, LM-14 and LM-15 are completed. AIUI the plan was to rebuild CSM-102, CSM-105 and possibly LM-2 as man rated spacecraft, but it wasn't done due to spending cuts from 1965 onwards and as the spending cuts weren't as severe as OTL or at least delayed ITTL they are rebuilt as originally planned IOTL.
Which leads me to the third piece of wishful thinking. Some of the money available due to no Vietnam War is spent on putting this hardware into space.
Apollos 18-to-20 aren't cancelled and due to some more wishful thinking Apollo 15 is still a J-series mission. Assuming that the 3 extra landing attempts were successful the number of successful landings is increased from 6 to 9. However, the amount of extra Lunar exploration is more than 50% due to the Apollo J missions being able to spend up to 78 hours on the lunar surface instead of up to 36 hours in the earlier missions, being able to do more & longer EVAs, carrying a LRV, being able to return a greater weight of samples and having CSMs with SIM bays. The number of ASLEPs left behind is increased from 5 to 8 and the Houston Control Centre which had been receiving the stream of information from them isn't closed on 1st October 1977. IOTL it was a purely economic measure, designed to save the $2M per year cost of the centre.
The number of missions to Skylab is increased from 3 to 8 and assuming that each of the 5 extra missions was of 84 days in duration increased the length of time the space station was occupied from 171 days to 591 days. Except, that the last 6 missions overlapped decreasing the length of time the space station was occupied by one or 2 months. However, the number of man days was still increased from from 513 to 1,773 days and due to missions 3 to 8 overlapping Skylab was a) continuously occupied for over a year and b) there were five occasions when 6 astronauts were aboard instead of the normal 3. ITTL missions 3 to 8 also delivered a Resupply Module (RM) to Skylab. The RM was proposed IOTL and was the basis of the Docking Module (DM) used in the ASTP mission.
Furthermore, the extra money allowed Skylab to run concurrently with the later Apollo lunar missions. Therefore, the Skylab OWS was launched in May 1972 or May 1971 instead of May 1973. The last of the 3 OTL missions to Skylab splashed down in February 1973 or February 1972. The 5 extra missions followed at intervals of 3 months so Mission 8 would splash down in May 1974 or May 1973.
IOTL NASA wanted to use an Apollo CSM with a SIM bay for ASTP to maximise the scientific return, but had to use an ordinary CSM for financial reasons. ITTL more money may be available, but NASA might still have to use a CSM without a SIM bay because all the CSMs with a SIM bay may have been used on Apollos 18 to 20.
More money may be spent on NERVA between 1965 and is OTL cancellation in January 1973 and ITTL the project may linger on until early 1975.
We still have the OTL Space Shuttle that everyone knows and loves, but approval for FSD might be given a year earlier, whether that leads to the first flight being in 1980 instead of 1981 is another matter.
Then it's 1975 and the OTL fall of Saigon and the end of military aid to South Vietnam so return to OTL expenditure on NASA.
That's at the very least.
At the very most the Apollo Applications Programme (AAP) is carried out in full. Though realistically it would be somewhere between. E.g. Skylab is launched in May 1971 but instead of ending with the splash down of Mission 8 in May 1973 the space station was visited by Apollo CSMs at 3-month intervals until Mission 16 splashed down in May 1975. In that case Skylab would have been occupied continuously for 3½ years from November 1971 (when Mission 3 arrived) to May 1975 (when Mission 16 left). In that case it was preceded by Mission 1 (the OTL Skylab 2) in May 1971 and Mission 2 (the OTL Skylab 3) in July 1971.
Edit: Some of that was written by
@Michel Van in Message 63 which I didn't see until after I wrote the above.