A few questions-
You know for sure the transmission worked fine before it was removed? Did you have any history with this vehicle to know any of its quirks?
You say when the transmission is hot it is slow to go into reverse or drive correct? But cold it is fine.
Is it only when you first select drive or reverse-or If you select reverse and feel it engage and go immediately to drive is it still slow engaging in drive or vice versa?
Does it engage faster if you are revving it slightly?
Does it engage in drive faster if you immediately set the shifter to low, not drive?
My best guess at this point is either a leaking accumulator or the low/reverse piston leaking-but much worse when hot. Both are known weak points-although the 42rle is really a poorly built and engineered piece
As for your transfer case I would change input and output bearings also along with the chain and anything else you see once disassembled. Your magnet looks crappy but you would be surprised how much accumulates on them. The big indicator is how big are the pieces. Slight sliver are ok but if there is anything bigger you may have something piled up.
Finally managed to get some time to answer this one after a battery replacement and a drive belt tensioner pulley failure.
I have no history with the other vehicle so no idea on the state of the gearbox before the swap, except what the mechanic has related to me, which with time has proven to be unreliable.
I can confirm when the transmission is hot it is slow to go into reverse or drive, But cold it is fine.
Once hot, this issue persists even when I going immediately to drive from reverse and vice versa?
When revving slightly, like around 1,500 RPM the delay disappears.
There is no difference at idle and hot if I immediately set the shifter to low, not drive?
Note that the Mercon oil has not being flushed out and changed yet.
I hope this clarifies and will help with any suggestions.
As for the transfer case, that issue seems to be heading to court at the moment along with the incorrect oil problem.