Although you will make the request for assistance with TUI, they pass your details to the airport as necessary. If you haven't done it already you will need to tell TUI what aids your partner will be taking onto the aircraft and what level of assistance is required, such as if she can manage steps and/or requires assistance to the gate, the aircraft door, or to her seat.
When you check in at the airline desk most airlines will afford you priority; e.g. if you are travelling economy you may be able to check in at the PE desk.
You will then be directed to the special assistance desk; they can offer your partner a wheelchair service to the gate or aircraft door if she requires that. If she doesn't then you make your way through security, using the priority or fast pass, or special assistance lane.
One thing to note at security is that her crutches will need to go through the X-ray. So, if she cannot manage without them you would be best to request the wheelchair service to at least the gate. The downside of that is that they won't take you on a detour of the shops and eateries, but you will generally get to the gate at lightning speed.
With regard to the last, I'm not sure if they would allow you to push the wheelchair and go off under your own steam; my wife uses her own chair so we haven't experienced that.
At the gate you will be one of the first to board; they generally hold everyone back until they receive word that you have.boarded.
Finally, you will hopefully have a pleasant flight.
Good luck!
