An international mission is currently not popular in the US.
China is way behind and rather slow while Russia doesn't have the money.
The Apollo program worked but was limited and under quite different political cucumstances. It was a US program alone.

But there are several advantages even under current problems:
An international manned space program would mean a treaty which would bind countries into achieving their commitments (that's why the Shuttle and the ISS programs are still active).
A well defined and advanced national program should be funded all the way, once a threshold is passed, otherwise it would be obvious to everyone who's trying to stall it and that would not be considered popular.
It's a lot a matter of pride.

I don't think many believe an international program without US help could get back to the Moon earlier than a US funded program.

But let's suppose it did: it would take time. Enough time for the US to see what's going on, convince the congress and catch up with it.