I'm guessing Husky or TTL version of it will probably not happen til at least March or April. Give time for the forces that fought in North Africa to rest and rebuild, make any changes necessary based on lessons learned, and bring in more forces to the region.
There's also the strategic situation to consider. The US and British have just had three major strategic victories back to back to back. A lot of stuff is going to have to be discussed in London and Washington.
Edit:
I think this was mention a while back, but I forgot, what battleships and aircraft carriers are the British currently working on. Given that the Germans and Italians have been gutted in terms of capital ships, I'm assuming a lot of planning on the Lion class battleships has likely stopped if it hasn't already been outright cancelled months ahead of time. Has HMS Vanguard even been laid down at this point? Given that the IJN has essentially been gutted, if Vanguard is under construction at this point, I could well see the Admiralty ordering construction haulted.
HMS Vanguard in OTL, by commissioning in 1946, had cost £11,530,503, including £3,186,868 spent on upgrading and improving the main armament.
https://en.wikipedia.org/wiki/HMS_Vanguard_(23)#Construction_and_career
Now I'm guessing at least some of that money will have likely already been spent, say 10% to 20%, but the British could still save several million Pounds by cancelling Vanguard if they haven't already.
Getting rid of the old R class battleships might be a good idea as well if they haven't already done so or otherwise assigned them to low intensity work. Outside of patrol work in the Indian Ocean, those ships never had good careers, and due to no interwar refits and upgrades, in many ways they where somewhat more of a liability then useful. Looking at scrapping them could free up thousands of personnel for other assignments, and perhaps save a fair amount of money going forward.