RFID reader sensitivity/read range - How does a "hard" ground increase sensitivity/read range?
Therefore I'd say, the decoupling is done porperly. However
  sensitivity aka read range is 5mm with the wall wart.


That assumes that the common mode AC voltage on the DC output of the wall wart is very low. It hardly ever is and the AC content is wiggling around at the low to mid hundreds of kHz. That's the switching frequency on the primary inside the wall wart leaking through to the secondary and wobbling both DC outs up and down together. It may be a few tens of milli volts but it can be enough to upset things.

This output common mode noise of the wall wart can be alleviated (and usually is) by having a class Y/X rated capacitor from the DC output to the rectified DC line voltage on the primary. Many suppliers of WWs use this technique but it comes with a problem.

That problem is a significant AC voltage at mains AC frequency on the DC output - it's harmless of course but, if the target load (your device) is not grounded then it's being shaken up and down by several tens (to hundreds) of volts p-p at mains frequency.

This will inject unbalancing currents into your load and potentially upset this or that. Hard to be definitive but it can (and does) happen.

So, my guess is that the decoupling isn't done properly OR your layout is a little flaky and is thus susceptible to unbalancing currents. Hanging a scope earth on the load 0 volts can be usually enough (in many cases) to solve this problem. Using a bench power supply that has probably got decent common mode (to ground) filters on the DC output can be enough too.

Of course, the wall wart might be just generating too much differential AC voltage at its DC output and this would be the place to review first. If it looks OK then use your o-scope and look at the WWs output voltage with respect to earth - each wire at a time and see what you see.