The only reason the Nords are suspicious of the Alikir is because they are extremely private
Source? And what do you mean by private? If you mean isolationist and xenophobic, that applies to the Crowns, traditionally. Post-Great War Hammerfell is uncertain, to say the least.
For those who might be not aware, Hammerfell consists of various independent city states and two main factions, and the Nords started a war with one of them back in the third era. What Bethesda will do with that is, of course, another question.
A lot of people seem to assume Hammerfell and an independent Skyrim would be friendly, but I don't think it'd be that simple. What did Skyrim do when Hammerfell was kicked out of the Empire and left alone to fight off the Dominion? Nothing. They just turned their head (metaphorically) and let it happen, like all the human provinces did. I don't think Hammerfell's going to be buddy-buddy with any of the human provinces for a while. Not openly hostile, but probably not friendly. “But why Hammerfell and Skyrim wouldn’t join forces against the Thalmor and the Empire? The enemy of my enemy is my friend, right?”. Wrong, very wrong. This alliance wouldn’t happen for 3 reasons:
1) First of all you're forgetting the Nord's great pride. It's part of what makes them WHO they are. They are not going to ask for help, which means it'd be up to Hammerfell (who already won their fight against the Thalmor) to approach Skyrim for help. And why would they if they don't need it? ON TOP OF THAT have you ever hear of a town called Dragonstar? It's a town that was half controlled by Skyrim, and half controlled by Hammerfell after the Nord INVADED AND TRIED TO TAKE OVER. The Redguards won’t just let Nords into their territory to "assist" unless they REALLY need it, which at present, they don't.
2) Do you think mutual hatred is enough to forge an alliance? Ulfric's Nords hate other races and the Redguards of Hammerfell don't care much for anyone at all. Just because they hate the same group of people doesn’t mean either of them is going to approach the other for help. And FYI, an alliance is defined by a mutual benefit, not hatred. Do YOU know how alliances between nations work? The hatred is obvious, but not enough to forge an alliance. When in Elder Scrolls lore has any alliance been formed based solely on mutual hatred? It's almost always based on benefit, and Hammerfell DOESN'T NEED anyone at present, they're doing fine on their own. Do you think Ulfric would send his Nords to die in the desert? Especially after losing so many during the civil war? Even if he could spare the men (he can't) what other benefit would he offer to Hammerfell? Tiber Septim had to support a losing side in a civil war, and win that civil war to have Hammerfell people allow themselves to be allied with the Empire. They don't just go around making pacts with people due to mutual hatred.
3) Jarl Balgruuf can tell you what Ulfric's motto is when it comes to forging an alliance: "Side with me or face my army", so no, Hammerfell/Skyrim alliance will never happen.
I think it's a bit of a mistake to apply individual human emotions to an entire nation, particularly one that's not very centralized. Two nations don't have to be best friends with benefits to see the need for a mutual protection pact or some sort of economic cooperation. If the need is strong enough, it'll happen.
It is... convoluted.
Using the Alik'r when referring to the Redguard is misleading as most of the Redguards do not live in the Alik'r Desert, although many who do are Crowns.
The general rule is the Redguards in the coastal city-states, save for Hegathe, are Forebears whereas those in the eastern region (Elinhir, Skaven, and Dragonstar) are Crowns, and much of the Kingdom of Sentinel consist of Crown fiefdoms despite the city of Sentinel being Forebear.
And the Alik'r is also home to tribes who are... something else.