Bowed neck?

I just bought a high end uke, and after playing with it for a while I realised that the neck is a bit bowed. If I hold a finger on the first fret and another finger on the 15 fret there's probably 0.5 mm gap (estimated) in the middle of the neck. It's a tenor scaled uke and the action is on the high side, about 3mm on the 12 fret. While it sounds good now, I'm worried that I will have issues when I lower the action.

What do you think? Should I go back to the shop?

It sounds as if you are describing what is called relief in the neck. Most guitars have some relief in the neck to allow the strings room to vibrate, typically in the range of 0.008" to 0.012" between the bottom of the string and the top of the 7th fret (when holding the string down at the first fret and at the body joint). Interestingly ukes tend to be built without relief, though I've never learned just why. In any case your 0.5 mm is close to 0.020" of relief, and that's way too much. It won't keep you from lowering the action at the 12th fret, but it will keep the action in the middle of the neck higher than it ought to or needs to be. I would take it back.

It would certainly be far more alarming if you put your fingers on those frets and there was no gap that you speak of.

A bit of relief (forward bow) is completely normal and necessary for proper tone and setup.
Ukes and guitars that have a truss rod are there to adjust the neck so it actually has a bit of that forward bow.

Perfectly straight neck with no relief would have horribly tone, playability and most likely some bad fret buzz.