Press the button, the camera takes the picture.
The time between those two events is called "shutter lag".

An old-fashioned film camera used mechanical linkages from that button to the actual shutter. But for the past 25 years, nearly all SLRs have had electronic linkages, film or digital.

Shutter lag is comprised of AF delay, AE delay, processing delay, and shutter actuation delay. These are all the same between film and digital, or at least should be all the same.

So film and digital ought to be pretty much identical here, on similar platforms.

Except that every 36th 37th frame (or less), there's a 1+ minute delay before a film camera can take the next shot. Digital cameras only hit this delay once every several-hundred (or several thousand) frames.

The next factor is shot-to-shot delay. Digital wins here, hands down: no mechanical delay while the film is wound to the next frame. The result is that digital cameras boast FPS (Frames per second) way in excess of similar film cameras.

And don't even get me started on shot-delay when changing ISO (film speed). At best, it's perhaps 5-10 seconds using film, when one switches from one camera body to another. At worst, we're talking minutes to change films. With digital, the delay can range from zero ("auto-ISO") to perhaps 5 seconds of fumbling with controls.

Cheers