Obviously with the Senate they can cut any budget/pass any law.

On their own the house can cut funding eventually. Push comes to shove, the house needs to vote for spending. They either pass a budget with the cuts, or the government shuts down.

There really isn’t a mechanism for the house that nationalize anything without the support of the Senate that I’m aware of. The house can call the vote day one to nationalize, but it’s symbolic without a legit path on the Senate