Historically the US was very keen on a 1943 invasion of France as has been discussed several times on this site. The British were very opposed and made two arguments against:
- We're not ready, the German Army is too tough and we should only engage them in geographically confined theatres where they can't bring their strength to bear because of logistics constraints (i.e. Southern Italy)
- All of our troops, shipping etc. is in the Med and we can't get everything north quickly enough post finishing operations in Tunisia to launch an invasion before the Channel gets too rough. So instead of losing the whole year we should go for Sicily and southern Italy and then moving the shipping north over winter
The US never really bought the first argument and dismissed it as the British being overly cautious but they couldn't argue with the second which is why the British strategy was followed in OTL.
Here both of those arguments are going to be weaker. The whole North Africa campaign is between 12-15 months ahead of schedule, depending on what happens with Vichy and it looks like by January 1943 there won't be any obvious amphibious targets left in the Med meaning that shipping will be free to go North in plenty of time for a May/June '43 invasion. Secondly the German Army looks a lot less intimidating both to the British because they've beaten them but also to the Americans because of prevailing racism. From an American perspective yes the Germans have beaten the French, Greeks and Russians but every time they've gone up against the British it's been an even fight and the Americans are certainly the equal of the British, therefore there's nothing to be scared of.
All of which means a 1943 invasion is very likely and for the command authority reasons I mentioned upthread I think it will be weighted very heavily towards US troops. With British troops being concentrated in different theatres (Italy, South France, possibly Norway) under British command.
The US Army did have a much better equipment acceptance process than the US Navy and it's stuff was generally very good but without the experience of North Africa and Sicily it's leadership, doctrine and training won't be and the blood price will be significant. But unless the US are willing to accept British leadership they won't be able to season their troops in the Med and considering 1917-18 and the lionisation of Pershing I think the likelihood is the US military, civilian leadership and public all accepting extra dead GI's as the price of keeping US troops under US command.