Well, I don't disagree with the propability of sinergy effects if one molds the EU forces into one but the AEW capabilities are hardly an example.
The 17 Sentrys at Geilenkirchen which are jointly operated by NATO are probably the most potent AEW force outside the US. If one adds the Sentrys of the UK and France we have a force of 28 AWACS in Europe compared to the 34 of the US (not counting the Navy Hawkeyes).
Quite a capable force and one of the few fields were Europe isn't lacking.
Don't forget that the 17 NATO E-3s are shared with the USA, Canada, & non-EU states Norway & (despite it getting its own B737 AEW) Turkey. There'd have to be some buying out or handing over of assets if they became purely European.
There are also several Swedish & Greek (again, despite it having a share of the E-3s) Erieye-equipped SAAB 340s & Embraer 145s in the EU.
One big lack is ground surveillance radar. The UK has 5 Sentinels, but that's not much.
The tanker fleet as a whole isn't too bad compared to anyone outside the USA (though tiny compared to that of the USA), but it's disproportionately British & French. The German, Italian, Spanish & Dutch fleets are small & diverse.
I'm with Stobiewan on the scope for sharing tanker, transport & MPA fleets. I'd also add sealift, where there is already some co-operation.