Sweden - is there a benefit to the Axis having Sweden as an ally? They were already selling the Germans all the raw materials they wanted. You might get a few divisions of Swedish troops to help back Finland against Leningrad, but is it enough to make a difference? Even if Leningrad did fall, what strategic difference does it make to the outcome of the war given what would later happen at Stalingrad and Kursk? Sweden joining the Axis would be like Ireland joining the allies.
Spain - well as we all know Spain was a nation that had just enjoyed 3 years of highly destructive civil warfare. Their infrastructure was buggered, they had an experienced but war weary and poorly equipped army, a navy vastly inferior to the RN, and plenty of republicans at home that would gladly cause problems for the regime with British backing. It is also very dependant on not only food imports but oil imports from the US, which would very quickly dry up and would add an additional drain to the stretched German supplies.
However they do have the strategic bonus of holding both sides of the entrance to the Med. Britain's hold over Gibraltar is tentative at best. The Germans had several plans throughout the war to pass through Spain and take Gibraltar. Even with the guns of the RN it's difficult to see Britain holding onto the rock long term. This loss coupled with the ever increasing difficulty of running the blockade into the med makes supplying Malta, Cyprus and the BEF in Egypt this way more difficult if not impossible in the long term, potentially jeopardising these.
Of course the flip side to this is Britain pushed Italy back across Africa largely because they were inept, and it took the Germans to reinforce them to balance the two armies. Germany isn't going to be able to spare any more forces once they've invaded the USSR, so it remains for Spain to back Italy up. Regardless, Britain can still back it's Egyptian army up via the India ocean and the Suez canal if necessary.
The canary islands would become important as a potential submarine base for the axis to raid British shipping, or for the the British as a key stop on the convoy route. There'll be a rush after Spanish entry between the Germans wishing to reinforce and the British wanting to take the islands, although given the topography they could be difficult to hold long term.
However as long as Britain can achieve the key objectives of holding the middle east and taking the Canaries, once America enters the war there's only one winner. Spain offers extra man power and resource but it also means more territory to defend, and Spain is a much better "soft underbelly of Europe" for the Allies to invade than the narrow, mountainous Italian peninsular.
As for Turkey, well they didn't have much of an army or infrastructure. Any additional forces moving into the Balkans or through the Caucasus in support of Axis invasions is going to be negligible, the former as it would fall anyway, the latter as it is mountainous with poor links leaning to long supply lines, and at the end of it is Stalingrad. Throwing a few hundred thousand Turkish troops into the slaughter is only likely to lead to early Turkish withdrawal.
The only real benefit for Turkey other than claiming some spoils from Greece is an invasion of the middle east. Aside from the issues that would arise with their purported allies Vichy France, the British forces in the region might well still prove too strong for them. Nonetheless the combination of Turkey attacking from the north, Italian forces attacking across north Africa back by both German and now Spanish forces, and the logistical issues caused by the loss of Gibraltar and the closing of the entrance to the med could lead to an Axis victory in the middle east.
Of course the strategic value of this is questionable - the chances of being able to drive through Iran into India are slim, supplying an army over that length of difficult terrain would be nearly impossible. Threatening Russia through the Caucasus makes little difference - the Germans still managed to take Stalingrad without this added route, holding the middle east wouldn't stop the Red Army encircling them and crushing the 6th Army. The real bonus would be oil, the Axis would have lots, and the impetus would be on the Allies to take the region back.
Long term I still see an allied victory. The Germans aren't going to be able to win a war of attrition against the USSR the same as Japan isn't going to win a war of attrition against the USA. Germany still needs to starve out Britain to knock them out of the war as they can't invade by sea, and even with the canaries this isn't that much more likely to happen.
R:
