The Azores were Portuguese, not Spanish. Unless the Brits opt to declare war on Portugal, they aren't getting them...
Let's see. During the war, the British:
- landed troops in Norway.
- bombed, bombarded and seized Vichy French naval assets wherever they wanted and seized French colonies whenever they could.
- occupied Iceland.
- replaced an unfriendly Iraqi government with a friendly one, not without waging a battle against the Iraqi army.
- bombed Finnish territory.
- collaborated with the Soviet occupation of Persia.
- occupied
Portuguese territory (Timor) and carried out a small military operation on
Portuguese territory (Goa).
All of that
without a declaration of war on the countries involved.
In some cases they had good justification (Norway; the Germans had systematically violated Norwegian neutrality first). In some others they had a shaky one (Iceland could be described as a Danish dependency and the Germans had occupied Denmark). In some others they had an acceptable factual justification, but not a legal one under international treaties and law (at Petsamo the place was brimming with German assets, in Goa there was a German interned ship acting as an intel and radio station). In some case there was no justification whatsoever.
Yeah, if British troops land in the Azores while Spain has gone Axis and there also are German troops in the Iberian peninsula, then the Axis might occupy metropolitan Portugal. Who would benefit from that, the Axis? Hell no, the Allies. It would be hard on the Portuguese, sure; but look at Yugoslavia. The British encouraged the Yugoslavians not to go Axis, then they could do little to actually help them; but it's not as if this resulted in a gain for the Axis.
But it won't come to that. The British won't immediately base ASW aircraft in the Azores. They will do exactly what they did in OTL, and the Portuguese will also do that. The British will outsource patrolling to the Portuguese, and the Portuguese will accept that. Maybe not immediately; in OTL, that happened in 1942, after the German declaration on the USA and the fact that at the gates of Moscow the German supermen had been shown to be such only in fair weather. But it will happen. Lajes will be turned into an air base, and Portuguese aircraft will begin doing the job. They won't attack surfaced U-Boote, true; but the important thing will be reporting them, as well as the weather flights. In case of emergency, Allied aircraft will be allowed to land there and refuel and be repaired.
Then by 1943, if not earlier, the Allies will be given full basing rights. As in OTL.