Wemos mini pro switch power from servo with mosfet bs170
My supposition:

The MOSFET's R(DSON) is too high and causing a large voltage drop.

A servo uses more current than a little LED.  That current passing through the MOSFET, with an R(DSON) of potentially up to 5Ω, reduces the voltage available to the servo, causing it to not work. Further more, ground for the servo is raised up above the real ground level adding an offset into the logic levels that could cause communication issues. 

Suggestion: Pick a MOSFET with a lower R(DSON) (ideally measured in the milli-ohms), and a lower V(GS) threshold (something around 1-2V max would be good).