Americans might say "I am going to empty the bath" or "I am going to drain the tub."

As far as the bathroom plumbing goes, flush usually applies to the toilet, although you can flush out your eyes in the bathroom, but this doesn't apply to the plumbing.

We would usually say "run a bath". This would be the process of filling the bath up with water. All the best, Chris

Drain

I would say to fill the tub and empty the tub. There is also the formal “draw a bath”. “Drain” is also fine - the tub is draining.