The inclusion of 1.a4 a5 doesn't alter much the evaluation of the position, but it will affect the choice of openings. Both sides' choices are reduced because the queenside is less flexible.
Probably White will want to play 2.e4 and not 2.d4.
Indeed, most openings after 1.d4 include a rapid c2-c4 to fight for the center, but here that move would come with a huge weakening of b4. If Black plays a classical Queen's Gambit or a Nimzo/Bogo-Indian system against 2.d4, the pin with ...Bb4 will be a serious issue; White will lack the ressource a2-a3 and all chances to fight for the initiative.
However, 2.d4 remains possible if instead of c2-c4 White plays systems with c2-c3, a move that correlates much better with a4. The London System and the Torre attack are totally legitimate choices after 1.a4 a5, and Black should choose carefully how to adapt to the extra queenside pawns moves. A kingside's fianchetto seems logical to me.
For similar reasons, the English Opening (2.d4, or 2.Nf3 with a quick c2-c4) shouldn't be played with White's pawn on a4.
2.e4 exploits the fact that squares c4 and b5 are more secure than usual for the Bf1. Black should adapt his response to the circumstances :
The Sicilian 2...c5?! is very unattractive : after White opens the center with 3.Nf3 and 4.d4 (4.Bb5 is reasonnable too but less ambitious), Black will lack his usual queenside play, and White can exploit the b5 hole. Svechnikovs and Najdorfs are ruled out altogether, the Dragon's queenside counterplay is hampered; Hedgehogs, Scheveningen and Paulsens might be the lesser evil but they are still unattractive with holes on the queenside's light squares.
1...e5 might not be the smartest either, unless you follow up with a Petroff. The inclusion of a4/a5 might slightly help White in the Italian, but it certainly does hurt Black in the Ruy Lopez. The Philidor is even more passive than usual with a closed queenside. In case of a Petroff, however, it is not obvious to me who is favored by the inclusion of a4/a5. There will be positional subtelties for sure, but it does seem like a decent choice for Black, and the best after 2.e4 e5.
Against Pirc/Modern White can play solid short castle systems and Black is deprived of his active ressource with ...b5. In the Scandinavian 4...Qa5 is just impossible and even after 4...Qd6 queenside castling will be riskier than usual. In the Alekhine defense, however, the pawn moves on the a-file help Black since kicking Nd5 with c2-c4 would only give it a superb outpost on b4.
Black most thematic choice, however, will be a light-squares strategy, the Caro-Kann or the French. ...Bb4 without fearing a2-a3 is a huge asset in the fight for the e4-square. After 1.a4 a5 2.e4 e6, I would not be surprised to see White prefer slow systems like the KIA (3.d3, then Nf3, g3 and eventually c3), or the advance variation 3.d4 d5 4.e5, inducing a weakening with ...c5 : 4...c5 5.Nf3 Nc6 6.Bb5.
The bottom line in every opening seems to be: White must choose variations that include c2-c3 or Bb5/Nb5 rather than lines with c2-c4. Black must prefer fight in the center over openings that count on queenside counterplay with ...c5 and/or ...b5. Quite logical.
I believe that in chess with 1.a4 a5 included, the evaluation of the starting position wouldn't change much but the choice of openings would. The most relevant would become the London System, the French, the Petroff, the Caro-Kann or the Alekhine defense. Others, like Queen's Gambits, and Sicilians would almost disappear.
What if instead of Chess960 we would roll a dice to decide include either a4/a5, or a3/a6, or h4/h5, or h3/h6, or nothing, at the beginning of each game ? With only 5 starting positions, it would not make opening theory disappear, but it would force each player to play a widest variety of openings...