Ramifications of P?NP Being Undecidable
In some sense, nothing much.  We still have P = NP.  It's possible for there to be statements that are true even though we can't prove them true.
In another sense, something interesting happens: in your scenario, we can explicitly write down a new algorithm that is guaranteed to solve 3SAT in polynomial time, using Levin universal search.  See https://cs.stackexchange.com/a/92095/755.  So, if there is a non-constructive proof that 3SAT is in P, then you can find a constructive proof that 3SAT is in P.
See also Is P decidable?, Assuming P = NP, how would one solve the graph coloring problem in polynomial time?, Explicit algorithms and algorithms involving unknowns.