Maybe

Let me go a little further.  Pump problem is typical reason for the delays
when cold.

It isnt the only reason.

Depending on the miles, and the frequency of use versus storage, it
sounds like you could possibly have piston seals that have hardened
and don't start sealing until they warm. By saying in all gears, do
you mean that the vehicle moves and delays shifting, or you have a
delayed engagement in all ranges? Typically, what I have described
would be a cause for delayed engagement. Used to be known as 'morning
sickness'.