We flew to Greece with them. Took a travel stroller that went in the over head lockers but others took normal buggys and checked them in at the gate to go into the hold.
https://www.tui.co.uk/destinations/info/travelling-with-infants-children#:~:text=Any%20prams%20or%20pushchairs%20need,for%20free%20on%20TUI%20flights.
We flew with Tui a week ago and took our travel pram in the overhead locker on the way out. On the way back though they made us put it in the hold. We were able to take it through the gate to the plane but once outside the plane they took it. Luck of the draw I guess! 😊
I think you get 2 items free of charge to go in the hold if you’re travelling with a baby x