What is the best practice and downtime for migrating an existing AWS environment to use the Transit Gateway?

You can accomplish this migration with zero downtime.

Steps:

1. Setup the TGW, attach the desired VPCs and configure the TGW route tables. (Don't update the VPC route tables yet)
2. Setup the central egress VPC to the Internet
3. (Optional) Create a test VPC, attach it to the TGW and test that the central egress functionality works as expected
4. Change the old VPC route tables and point 0.0.0.0/0 to the TGW instead of wherever it points today. The trick here is that you cannot create a second 0.0.0.0/0 entry. (Therefore create 0.0.0.0/1 and 128.0.0.0/1 and point them both to the TGW, then delete the old 0.0.0.0/0 route, create the new 0.0.0.0/0 entry and last but not least delete the two x/1 entries.)
5. (Optional) Remove any VPC peering that you no longer need.