Bought a 240d about a year ago with 220,000 miles.
Had a delayed reverse, 2 or 3 seconds. Changed tranny fluid, AMSOIL, same delay. Figured I would just put up with it. Added another 25,000 miles in the past year.

About a week ago the delay suddenly started getting much longer, 10 seconds or more but it was fine after car was thoroughly warmed up.

I have a 250 foot long, steep driveway. In neutral I can do 20 MPH by the time I hit the street. What I always did was drive forward up the driveway to park at night and put it in reverse for about 2 seconds and keep my foot on the brake approaching the street backwards.

Not wanting to be stranded at the top of the hill I started backing into the driveway at night.
A few days passed and I noticed the delay getting shorter. As of today there is no delay at all, a solid reverse. All I can guess was something blocking the valve body.

Anyway, I'm sure a lot of you have a delay in reverse, you may want to find a steep hill and back up it a few times to see if it helps.

i may have to try this as mine has similar symptoms to yours