So what was the fallout between House and The Enclave that puts them at odds over the NCR?

This assumes them working together on certain projects means they are aligned in their goals. Look at the real world and see how often governments and other governments or governments and NGOs/corporations cooperate on one project whilst working against each other on another.

House worked with Vault Tec and the government because it was in his interest. If he felt the NCR threatened his interests he'd probably decide to work against them. In New Vegas he wants the NCR stable enough to function sort of how the US functions for China - As a consumer to whom one can offload extra capacity and extract wealth.
I do not know why he'd want to nuke the NCR though, it doesn't make sense. So either it wasn't him or it's bad writing or a good reason will show itself.