If you are under the same lighting conditions and have the same settings selected when manually pushing the shutter button each time (the flash always fires) as you do when using the self timer (the flash only fires once), then it appears that the limitation of only the first frame firing the flash when using the internal flash (see pages 70 and 148 of the D7200 User Manual) also applies to an external flash when using the self timer. Page 70 also seems to indicate that only one frame will be exposed. YMMV.
The instructions for Interval Timer Photography on page 217 of the D7200 User Manual specifically exclude using the self-timer with the interval timer feature.
Select a release mode other than self timer and Mup when using the
There is a built in 3-second delay when pressing the shutter button to start the interval timer when Now is selected in the Start Options (p.220) of the Interval Timer menu.
Your D7200 also has an option for Time Lapse Photography (p.171) which does seem to allow the flash to fire with every frame (p.176).