Unfortunately, you have a very common problem in the older or “high mileage” 4T60E transmission. It is normally described in the transmission shop as; "Morning Sickness". The problem, is the internal lip seal to the input clutch in the input housing. As they get old, they shrink and get hard, therefore, they do not seal to the housing the way they should. This problem will cause the input clutch pressure to bleed off and not apply the clutches. Once the fluid starts to warm up, the seal expands enough to seal and will usually be fine until the fluid is cold again. You can get an additive at the dealer similar to “Sea Foam” that will soften the seals and will work for a while; however, this is not the long-term cure!
The only long-term fix, is to remove and replace the seals. The Trans will have to be removed and disassembled. Better to let a pro do this, since special tools and equipment are needed.
Here is a trick that I normally suggest, to help overcome this delay. The reverse and low clutch assembly work together and if your delay is in reverse, start the car in park and immediately shift to low gear (1st) until the car moves, then back to reverse. The opposite applies if the delay is in drive, (usually happens more often than reverse). Start the car, then shift to reverse first, wait a few seconds until you feel it engage, then shift to drive and you are own your way. Again, this is not a long- term cure; however, it will keep you going for awhile until the weather gets above 50 deg. By the way, 50 deg. + or - 5, seems to be the ambient temperature that this problem will appear.