I'm looking for some rule-of-thumb guidance on this matter. It relates to route 4, the westerly take off from London Gatwick (LGW).
While in theory possibly a duplicate of the other question, and while the physics are very well explained there, the numbers to be used for takeoff and landing aren’t given in the other thread.
The following works as a rule of thumb for most commercial airliners: Maximum bank angle after takeoff with all engines would be around 25 degrees, with a failed engine it would be around 15 degrees until minimum clean speed is reached and the flaps are up and 25 degrees after that.
True air speed shortly after takeoff shouldn’t be much above 200 knots for most weights, airfield elevations and temperatures encountered except in extreme circumstances.
Plugging both these into the formula for turn radius explained elsewhere gives just above 2300m radius on all engines, and just above 4000m radius with a failed engine and before cleanup.
Brief answer, but I hope it helps!
You could look to the Standard Instrument Departure (SID) routes published by NATS for a guide as to what is done - as opposed to what could be achieved.