In terms of repair time I need to decide if bomber command manages to hit any of the ships in yard hands. Graf Zeppelin will be ready soonest at 3 months, the Tirpitz a month later at 4 months and then Scharnhorst at 6 months.
Once they refuel and rearm Bismarck she could head out again at any time.
The Germans might want to want to try to send Graf Zeppelin out with Friedrich der Grosse in Ausgust / September. Assuming the Luftwaffe releases the aircraft that is.
So, if they plan a sortie in October...The Kriegsmarine would have
Bismarck
Tirpitz
Friedrich der Grosse
Gneisenau
Graf Zeppelin
The RN can have
Duke of York
Repulse
Renown
Rodney (Maybe; her refit may be delayed ITTL due to capital ship shortages)
Nelson
Barham
Valiant (Could use some time in the yards)
Malaya
Revenge (Possibly; she's going to spend some time in the yards, and may not be done)
Ramillies
Royal Sovereign
Obviously, the RN must also stock the Med. I would see Repulse and Renown along with Ramillies and Royal Sovereign as most likely candidates. Hood's quick death will discourage the use of Repulse and Renown near the Germans, and the R class are too slow. Revenge may not be available yet, she has a decent amount of damage to patch up.
Thus, the Queens would be brought back to Scapa as Home Fleet. Rodney may have her June refit delayed due to the severe shortage of ships. The RN will also have no shortage of carriers. Ark Royal and Victorious need repairs, and may not be ready in time. Regardless, not less than two carriers being part of the Home Fleet.