Presuppose there is no Operation Torch, for any of a variety of reasons.* The allies never invade French North Africa from the Atlantic of western Mediterranean sides. (so no Operation Gymnast landing in Morrocco either).
What do the US/UK do instead? The most likely alternate plan is maximum reinforcement of 8th Army. This is awkward, because of the distance, but it could work.
I'll put forth an initial hypothesis and set of assumptions you can feel free to reject or modify as you wish:
First I assume-
A) Regardless of the lack of Operation Torch, the 8th Army is going to drive the Afrika Korps all the way out of Libya in the month or two after El Alamein.
Presumably Second El Alamein, i.e. Operation SUPERCHARGE. BTW it took Eighth Army three months to reach Tripoli from the start of SUPERCHARGE (10/23/42 - 1/23/43).
The 8th Army was built up to unprecedented strength by this point and Rommel was at the end of his logistical rope.
Fair enough. And with no TORCH, Eighth Army will be built up faster and attack sooner.
The Deutsches Afrika Korps (DAK) was only a small part of the Panzer Armee Afrika (PAA) under Rommel's command.B) The Germans will neither withdraw Afrika Korps...
nor allow it to surrender or allow it be overrun without committing considerably more effort to rescue it.
Not clear. There was no commitment to reinforce PAA after Second Alamein that I know of. Hitler responded to TORCH by occupying Tunisia.
From there I hypothesize-
The Germans will probably send forces to occupy French North Africa to give Rommel a line of retreat and/or secure base as he is pushed from Libya.
Extremely dubious. Such a move would drive all French colonies over to the Allies. The Germans could not occupy Morocco or even Algeria before the Allies could get troops there. And with Rommel in full retreat, being driven out of Libya, the French commanders in North Africa are not likely to roll over for the Axis. Opportunists jump on winning bandwagons, not losing bandwagons.
The French in North Africa were divided. The main body wanted to remain neutral and avoid any fighting. They would oppose Axis invasion, but some might not dare to resist (such as Esteva in Tunisia). Others were covertly pro-Allied, and were organizing for a coup d'état in Algeria.
If Axis forces invaded French North Africa without US/UK provocation, most of the neutralists would join the pro-Allied side, and there would be substantial resistance to the Axis. Within a day Allied forces would be on the ground in Morocco and Algeria (by air; in a week by sea).
Axis movements into North Africa would be far more difficult than OTL. One assumes that the Axis does not invade Tunisia until 8th Army is closing in on Tripoli. By that time, the Allies will have easy communication with Malta, and built up air and sea power there to interdict Axis communications to Africa.
(OTL, the Axis move was in November 1942, with Eighth Army still deep in Egypt, First Army newly landed in Morocco and Algeria, and Malta still under siege.)
Considering these factors, IMO Hitler would have lost any taste for further African ventures. So the remnants of the PAA would be evacuated. The French would probably agree to letting PAA's personnel pass through French territory, and maybe even some equipment, but only for transit home, probably in January 1943.
In the next several weeks there would be a complicated dance, as the Allies conspired with the pro-Allied faction in Algeria (who were not affiliated with De Gaulle and Free France). De Gaulle would try to get involved. Opportunists in French Africa would skew toward the victorious Allies. At some point in early 1943 there would be a coup d'état combined with a US/UK landing and invasion (from Libya). That's what TORCH was supposed to be, but the coupsters weren't ready. Three months later, with the Allies much more strongly posted, there won't be any resistance.
So overall, the Allied victory in North Africa is moved up by three months; but Axis losses are far smaller. (No 275,000 prisoners in Tunisia.)
The Allies invade Sicily in early May. There will be more Germans and front-line Italian troops there, and First Army will be almost entirely green. So the campaign lasts twice as long as OTL