Either way. I'm not too well versed on exactly when Spain and Portugal first grabbed them, but I'd assume that's between 1400 and 1600 at least. Perhaps Sir Francis Drake could claim them? Or at the defeat of the Spanish Armada, claim the islands?
In that case the Azores could potentially be colonized by someone else, you just have to have Portugal not be as interested in them, thus leading t no one officially claiming them awhile longer, though even then, aide frm Portugal only Castille and England I think would do so.
That said, it's more likely that any non-Iberian country controlling them would gain control via taking them from Portugal and Castille/Spain.
The Azroes I don't really see anyone but England/Britain being interested enough to try to get them, though I suppose, before a certain point, you could see a treaty where various colonies are being traded and the Azores ending-up being given to some country as compensation, most likely France.
The Canaries I could see France or England/Britain trying to get in a peace treaty with a Spain on the losing side, I suppose maybe you could see Germany gaining them following a devestating war for Spain (whether Germany was involved or not); I seem to recall reading something about the U.S. originally planning on trying to capture and take the Canaries during the Spanish-American War.
While much less likely the Dutch might take them (they sent a large amount of troops and ships during the Dutch War of Independence), and at various times Ottoman-aligned pirates captured some of the islands.
Madeira is a case where you could see it colonized by someone else, most likely England/Britain or France, though on an off chance perhaps Genoa or the Dutch; IOTL the islands were only colonized after a Portuguese ship was blown off course in a storm and landed there in desperation.