Here is my experience and solution for no transmission engagement after a long cold soak of my E46
https://xoutpost.com/1150737-post188.html
IMHO, if all other warm temp ops are normal and you have to wait a while for coolant temps to rise to near normal operating before normal trans engagement, then you might consider changing out both of those solenoids. It is a relatively easy DIY and the solenoids are not that pricey if sourced through low cost sources.
Mike