Vehicle Issue Description (Automatic Transmission): When I start the car in the morning (cold start), a warning message (Transmission Limited Function check manual) appears briefly on the dashboard and then disappears. While driving: The automatic transmission shifts into 2nd gear It then will not shift to higher gears. The car feels restricted, as if it is in limp mode After driving for about 1–2 miles, if I: Switch the engine off Then switch it on again The car then drives normally, and: All gears change smoothly as expected No further issue for that drive This problem mainly occurs on cold starts and temporarily clears after restarting the engine.

Welcome, I'll start...
THIS IS CRITICAL...When was the fluid and filter last changed (if no record, or even if there is, just change it, look at the oil, tear the filter apart to check for plastic or other debris)?
If that doesn't work or even if it does...
Get Forscan and Bluetooth or USB lead, get the exact fault codes.
Always do everything possible external (including wiring integrity from PCM to transmission etc) before even thinking about any v. expensive internal work which we hope isn't necessary, yet.
As, you say, as it appears temperature related and works OK once warmed up, I'd love to hear the oil change history!
I suggest (as usual):
https://www.ebay.co.uk/itm/112436788478
BUT, you know there's a VERY specific process that has to be followed, don't you?
https://www.youtube.com/watch?v=CERqRSK2SWM
Or this for DIY but I do it by setting each wheel on a block (or bricks) to raise it a little to get access whilst keeping it level:
https://www.youtube.com/watch?v=EDmo9Qv0g0Y