My first thoughts on seeing the subject is that it has got valuable resources, as often discussed (and disputed, it has to be said) at other times. 3He is a little futuristic for us right now to look at processing in bulk (certainly was in 1969) but perhaps when we set up the moon-base we can do more with it. The mineral wealth (known and unknown) in the regolith probably isn't worth the equivalent costs of trans-shipment to Earth except for something really exotic, but when it comes to building Moon-side or even Earth-orbital items then (prior to decent economic exploitation of asteroids, and getting them or their processed materials back into a Sol-3 orbit) it'll be surely cheaper to source from the Moon than launch large amounts of the same stuff from Earth (no matter how ubiquitous and already infrastructure-ready equivalent Earthside resources might be).
Which is not to say it'd be easy. And you wouldn't use a Saturn V styled mission to go out and retrieve moon-materials one LM's-worth at a time, or even slightly more just to Lunar orbit, there to be bunched together and then sent (unmanned) in bulk to LEO/wherever with a more efficient but slower trajectory.
For a start, we'd be sending people out on extended missions (possibly indeterminately so, at some point). Moon surface (and sub-surface) accommodation, automated supply-ships sent for what can't (eventually) be self-provided with a hunk of pre-sent equipment, and all that jazz. That includes at least rudimentary (sufficient, if not the best that would be made here on Earth) propellant manufactured on the Moon, if not quickly progress to Linear Accelerator launching technology for (unmanned) loads. Space-tugs (automated?) of some kind would rendezvous with the projectile somewhere near the top of its trajectory (probably set so that a miss merely means crashing back down on some currently 'safe' part of the Moon[2]) and swing it into a holding orbit ready for sending on to 'wherever', as already mentioned.
Much as I liked the Shuttle, the post-Shuttle re-emphasis to separate passengers from cargo, when it comes to lifting, is probably a wiser decision than making a combined transport 'cargo-bus' system. Add to that staging the travel arrangements (a reliable to-orbit and from-orbit vehicle, of whatever form, only being used to rendezvous at some way-station with the trans-orbital craft that has been assembled up there with the intention of remaining space-bound, for example) then we have a system we can probably rely upon at least until someone gets something novel (space elevator/fountain/tether-rotors or even by a massively unimaginative leap of imagination, By The Great Montgomery, 'transporter' technology itself!) sorted out as a replacement.
Anyway, this is a possible "now and future" answer to "What will happen...".
If the question is more about what would have happened in 1969, or thereabouts, "...if the Moon had had extraordinarily valuable resources", then the race for the Moon would have not been 'won' by the first exploratory landings, but Russia would doubtless have redoubled their efforts to get there (second, but now with a 'claim', however much this would be against the extant Space Treaties) even while the US found every excuse it could to not make their own missions an "International Moon-station" mission, in order to at least forestall the competition until "Moon-Mir" got established to mirror the work of their own "Moonlab"[1].
Where it went from there is anyone's guess. Depends on if anyone ran out of money (or "leveragable political power", in the case of either party) in getting this set up... on any 'accidents' (or accidents, unquoted) suffered by either side... on what the Chinese/Indians/heck, even us British (maybe or maybe not separate from the Europeans, or choice European powers[3]) then did to get (back) into the race.
Apologies, BTW, if this post seems disconnected. I'm currently running off a portable version of a browser, for various reasons, and on this hardware (maybe USB 1.1-only ports, and the browser caching the input through it near-continuously?) I'm getting severe pause/slowdown. Type several words, wait for them to appear a few seconds later. (And don't ask me about the difficulty of going back and editing and correcting typos/thinkos!) Really disjoints my thinking/writing process. I have lots more to say, but (lucky you) I'm cutting it off here, after spending half an hour, I think, on just this brief version, and I shall forgo even the usual Preview (though doubtless I;ll have the "n posts since started editing" message to deal with). Thus "E&OE" is a definite caveat to what I have writ.
[1] Considered "Lunalab", instead, but doesn't work in my mind.
[2] Which, in a generation or two, as the technology becomes passée enough, becomes the location of Scavenger parties. Independent Lunar traders who can make a quick spacebuck from finding and retrieving lumps of refined material that were never caught.
[3] Whether as in Concorde or explicitly not in concord!