Very few airlines will transfer luggage to a different airline on a separate ticket, and when it happens it is mostly due to the agent not realising they should do so.

Neither Easyjet nor TUI will transfer your luggage between them, in either direction.

Easyjet will only transfer luggage when the Easyjet sector has been booked as part of a single ticket with the very few airlines with which Easyjet cooperates for these purposes.

But there is no chance of U2 transferring luggage to TUI. Plan accordingly - you will have to wait to collect your bag, go landside, and observe the check-in deadlines of your second ticket. Be aware of the possibility of delays or loss of baggage (and the need to report the loss at the airport before departing for your second flight).

You should look into the Gatwick Connects service.