I recently bought a book, the "Lunar Exploration Scrapbook," which includes CGI and explanations of all variants of this idea. If I had a scanner, I could show them, but I'll have to settle for typing out the descriptions. Just give me some time to find it.
Ah, that's a cool book. I got it for my brother a while back.
EDIT: The POD for this might have to be in the late 1950s, though. Project Apollo, a 3-man spacecraft for LEO and circumlunar applications, was NASA's goal since Eisenhower, and was intended to be the direct follow-on to Mercury. When Kennedy announced the Lunar goal, the work done on the Apollo Command and Service Modules was just redirected as a lunar ship.
Yes, a better POD might be during the Eisenhower administration; have him be a bit more favorable towards space flight, choose the Army proposal for a space launch vehicle instead of the OTL Navy-Vanguard proposal, and you might be a year or two ahead. This probably wouldn't affect the 1960 election much, so you could still plausibly get Kennedy in. Have the Russians still launch first (hard, but possible), and he still does the "land on the Moon" thing; the extra year or so of serious development, combined with no Apollo fire, means that the first landing could take place in late 1968 or early 1969. Maybe even early 1968.