The 12th Amendment clarifies that it must happen after the electors have been formally counted, so it would definitely be the new congress. That means a chance of a Democratic VP, but unless Bloomberg has an army of 100+ indy House candidates in the waiting, Republicans are still nearly certain to control 26 state delegations, even in a situation where Democrats narrowly won the House majority.
That could mean Trump as President and Warren as VP - let the fun times roll.
Of course in reality, that would not happen. Bloomberg could lobby both GOP and Dem moderates so anything is possible! Not to mention that for all this to transpire, a lot of things have to go right (or wrong
).