This is most likely a camera behavior, not a recharge issue. On the Nikon D7200, self-timer/continuous release behavior limits flash firing to the first frame. The manual indicates that in these modes, the built-in flash only fires once, and community reports suggest the same limitation can apply with an external TTL flash during a self-timer sequence.

So if the flash fires normally when you press the shutter for each shot manually, but only once during the self-timer burst, that points to a D7200 self-timer limitation rather than a problem with your Neewer flash.

If you need flash on every frame, the practical workaround is to trigger each exposure separately rather than using the multi-shot self-timer sequence. You may also want to look at other release methods or interval shooting options in the manual, but Nikon specifically separates interval timer shooting from self-timer mode.