Connecting two LED stripts with individual power sources to one MOSFET
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. OP's scheme.


  Would I need any protection (diode?) to prevent current from flowing from one supply into the other? 


You have identified one problem and you are probably correct in your thinking that, with both supplies on, the LEDs themselves would prevent backfeed in the event that one supply was a little higher than the other.

You also have to consider other potential failure modes:


If one supply fails you will have a long-term 24 V reverse bias on that chain of LEDs.
At start-up and shut-down the supplies will rise and fall at different rates. This won't be as bad as a permanent difference.


A separate diode in each LED string leg might give some added peace of mind at little cost.

Of more concern - and I don't know the answer - is that the Ikea power supplies are switched mode and you are adding a PWM load to them. I would imagine that the PSUs are optimised for their designed steady load with the minimum component count, etc., to achieve that reliably. By using PWM control of the load there may be a chance that you affect the stability of the PSU. You might get away with it forever, for a time or not at all depending on whether or not you are close to the switching frequency. If you're fortunate the PSU will shut down. If not it might overheat.

Others may have more experience with this.


  Also, assuming each strip consumes 44mA at 24V, how much current and voltage would pass trough the MOSFET?


Peak current will be the sum of both, 88 mA.