Oxbridge suffered a lot of collateral damage from Brexit because of EU funding cuts and massive loss of EU staff and EU students, who now have to pay foreign fees (4-5x regular home fees). An increase in fees also made it prohibitively expensive to hire EU PhD students.

The situation is slowly recovering, as the UK has now first-class access to EU funding programs and there is an open negotiation to bring back home fees for EU students. However, visas are becoming more restrictive and the exceptionally high fees associated with them might be again increasing, which is putting off potential new employees.

Besides, I am not sure Oxbridge has sufficient extra spots for overseas students diverted from the US due to its peculiar tutorial system. There are lots of top EU universities that could collectively benefit from this as they are much cheaper and larger: Heidelberg, TUM, KU, DTU, KI, KTH, etc.

Harvard is liberal arts, so maybe Oxford.