In order for "the Axis takes Gibraltar" to work the POD needs to be Franco wins the Civil War in less than a year. By the end of the Civil War Spain's economy and infrastructure, which was backward even before the war, was in terrible shape. Just transporting and keeping supplied the German forces listed in an earlier post would be difficult if not impossible and certainly would take some time giving the Brits plenty of notice. This gives them the opportunity to act preemptively and hot areas of concentration of the artillery and transportation systems. Because of the limits of the Gib airfield these might be shuttle runs, and won't last long but will make life difficult for the Axis. Also because the preparations will be so transparent the Brits will be able to evacuate many non-combatants & bring in supplies.
Even if the Civil War is very short, and Spain not devastated, essentially all oil for Spain was imported, much from the US - join the Axis & that stops.
BTW in terms of the Canaries, the Germans could have transported a division or 2 there as well as some Luftwaffe assets covertly or semi so in Spanish flagged vessels before the attack on Gibraltar. With decent troops, some air support in the Canaries, the Brits can stage a naval raid but an amphibious landing in 1940 is very very iffy. The Kriegsmarine would have loved owning the Canaries as an advanced base for subs and recon a/c.
However, the only way Franco would join Hitler would be if he had won very rapidly in the Civil War and therefore Spain was in decent shape AND he thought Hitler was going to win AND he was offered Gibraltar & choices bits of the French Empire.
After the meeting at Hendaye in 1940 or 1941, Hitler described dealing with Franco as being like going to the dentist.