Opinion-based, but I'll give it a shot.
In the timeframe of one year, it's not possible to develop any new launch technology of consequence; we have to rely on existing launchers, with a couple of possible exceptions for launchers that are well in development.
SpaceX has the best track record for high production rates and rapid scaling, so if I were in charge of the budget allocation, I'd put most of the money and attention on the existing Falcon 9 or Falcon Heavy. They're a reliable launcher; the first stage (and FH boosters) are reusable, so we don't have to build as many. user2702772 points out that FH gets more payload to LEO per expended second stage, but F9 might be flyable from more launch sites, so I'd guess we'd want some mix of the two. SpaceX isn't set up for a lot of concurrent flights at present, so improving launch site logistics might be as important an investment as building additional stages. We'd want to fly RTLS missions instead of barge landing missions -- less upmass per flight, but much faster turnarounds. The important thing here is that we give SpaceX the money and let them figure out how best to apply it without too much micromanagement. I'm not the biggest fan of SpaceX, but agile project development is what they're good at.
Second priority is to go to everyone else -- ULA, Roscosmos, China, ESA, ISRO -- and ask them how many rockets they can build in the next year and how much that'll cost, give them twice as much money as that, tell them to do their best, and then forget about them. Those agencies aren't set up to change their plans quickly, so I wouldn't expect too much from them.
Starship/Super Heavy and SLS are potentially useful in this scenario, and Starship/SH in particular can be relatively rapidly built, apparently, but the ultimate limit here is how much manpower can be brought into play to construct, stack, and manage the rockets. I'd give Boeing et al as much money as they thought they could spend -- one SLS launch provides something like 8 times as much mass to LEO as a RTLS Falcon 9. If SpaceX felt they could get Starship/SH flying, more power (and budget) to them, but my guess is that, logistically, it will be easier to snap up additional factory space that can accommodate manufacture of the smaller Falcon 9.
For a 5-or-more-year plan, I agree with Slarty that Starship is the smart money, but start building Falcons anyway in case the design turns out to have issues that take years to resolve. Even for a 10 year plan, I don't think Sea Dragon is the strategy to pursue; there are too many unknowns in building a booster that size.