It would take a year to let contracts, and another two years to get some hardware delivered. At the minimum, three years. Probably more like 5 - 10 years, which was the original schedule in the 60s.
Provided you got a serious, no-BS management team -- which is what's missing from just about every space program I'm aware of. With the right management team, it would go pretty quickly. With the wrong management team, it would never go at all.
But you're right. We have the basic elements of heavy-lift capability with the Shuttle system, which can toss 250,000 lb into LEO. NASA explored the Shuttle-C concept many years back -- replace the Shuttle with an engine-holder and a big empty cargo cannister.
As I recall, there were no big technical hurdles; rather, they determined that there was no need for a booster that big. They were right, too.