Bird strike, engine surging, severe vibration, flames out the back and 131 says don't declare a Mayday. I don't agree. The best course of action in the case of the TOM was to declare a Mayday - particularly in the first instance. This is a shared view between professional pilots and ATC as demonstrated in this thread.
131, there's a significant difference between private flying with only your good self aboard and flying around with 200+ behind you and in a large jet with the potential to do a remarkable amount of damage. This isn't pontificating; merely suggesting that if you were in command of "heavy metal" that you'd take it as seriously as those of us that do it for real. The TOM example was an excellent demonstration of how to do it well.
Anything other than a normal landing flap at below maximum landing weight is a special case. For single engine operation the 757 lands with reduced flap, has increased landing speed and distance and the brake energy is greater than normal. Remember that there is no fuel jettison system and the flight had just taken off and had a fuel load appropriate to its planned trip - i.e. heavier than normal landing weight. At commercial jet weights, that increase in energy and directly, brake energy, is significant. It is a sensible precaution to have the fire services check the landing gear, brakes and airframe for damage. Remember that in most cases, this will be the first opportunity for anyone to actually see the damage sustained since the view from the flightdeck and cabin will present very limited information. Most TOM 757s do not have brake temperature indications fitted. Fan blades have an enormous energy and if uncontained, can cause significant damage. An inspection is vital after a landing of this nature in order to determine what the safest option is in respect of whether to evacuate, disembark prior to stand or to continue to stand.
The reason for describing what radial they were flying is due to aircraft performance and obstacle clearance with an engine failure. Each operator determines their own engine failure procedure and these are not known to ATC since they rarely match the SID being flown owing to a significant decrease in climb gradient. The procedures also change from time to time. The aircraft must accurately fly this manoeuvre and with the traffic density in UK airspace, knowledge of the intended path of the aircraft is vital to ATC.
Markedly increasing rotation rate to miss a bird (when it will already have passed under the nose and will no longer be visible) is mistaken at best and at worst will lead to an over-rotation, tail-strike, and compromised initial climb performance and likely, an airspeed below V2, just when you most need it having, like in this case, lost an engine.
Passenger safety is the responsibility of the commander and as I've already stated, the importance of that must never be underestimated.