AHC: Commercial lunar tourism by the early 2010s

It might well have to be pre1900....

Doesn't that seem a little too pessimistic? I mean, there have been plans to establish a military base since the 1960s and a scientific research base since not too long after that...what's to say private enterprise couldn't accomplish it? Some estimates place us having regular space tourism within the next decade, so to me it seems if a few alternate events had been set in motion that significantly reduce the cost of manned transport, even if it's for the rich, it isn't too far fetched.
Avoiding WWII is one approach. The sheer human and economic costs of that war constrained rocket technology to a lot fewer players than would have done it without the war, IMO. Imagine if the engineers at Avia or PZL or IAR got to work on their own missile and jet designs rather than license-building Soviet designs. Imagine if millions of potential Eastern European and Chinese and Japanese engineers weren't slaughtered in the fighting and bombing and gas chambers.

But if we stick to the period generally defined as "the space age," that is, After 1945 (when scientific use of V-2 rockets began)...

You want to reduce launch costs, and that probably means getting an economical reusable rocket by 1990. The USAF's Aerospaceplane project had some promising ways to achieve two-stage reusability, but their obsession with SSTO hamstrung their efforts. If they aim a bit lower, going for a reusable two-stage vehicle, they might get at least one stage flying in the 1970s in lieu of the Space Shuttle, and then a reusable second stage can follow in the 1980s. Once you have cheap access to Low Earth Orbit, moderately-priced lunar orbit excursions are plausible enough--you could get Low Lunar Orbit tourism by 2000. The landing is tougher, since that requires an extra vehicle, but that might be something in development by today.