How would the Royal Navy pursue Prinz Eugen?

In the alternate timeline where Rodney and the Tribal-class destroyers sank Bismarck, would the Royal Navy then hunt Prinz Eugen?

Yes. The Royal Navy would continue trying to find and catch Prinz Eugen.

Sinking Bismarck in the hypothetical encounter would leave Prinz Eugen as a further target. The Royal Navy would still try to find and catch her.