Under Public Transport rules, there are strict and clear calculations which you must make with regard to how long a runway needs to be. They revolve around the need to cope with an emergency such as an engine failure.
Having started your take off run, you need to be able to stop, without damaging the aircraft, within the runway length plus any stopway at any speed up to V1. (V1 is the maximum Indicated Air Speed you can reach and still be able to stop in the remaining distance). Beyond V1 you are committed to continuing to rotate speed (VR), (even if an engine fails) and a subsequent take off and there are rules about obstacle clearance following an engine failure. VR is a speed governed by the weight of the aircraft and its flap setting, not the length of the runway. The lighter the aircraft the less lift the wings need to provide and VR will be slower on a lighter aircraft.
So, there is a gap between V1 and VR. The size of this gap is also regulated for obvious reasons. Say you decide that you can get to 80 kts and still stop by the end of the runway but that you cannot rotate until 120 kts, there is a 40-kt no-mans land when you can't stop but you can't fly either. This is not a good idea!
What all this means is that it all depends upon runway lengths (or, more correctly, Take Off Distance Available-TODA) and aircraft weight. So arguably it is perfectly possible for a light 777 to take off from Hawarden given the right criteria. Only someone with the performance charts for the aircraft could give you a definitive answer. Don't forget Air Force One B747 departed Teesside (2291m) for the US direct the other day. The TOW for a B747 was probably quite light and I'm sure they wouldn't risk it if it wasn't within limits!
In all of this, I have only considered the take-off scenario. There are also limitations on landing. But there is no point in landing at an airfield which you can't then take off from (unless you happen to be flying Concorde last week..
With regard to your main question: It is the Captain's ultimate decision. All an ATCO is do is give the relevant warnings but you cannot physically stop them landing. They can expect lots of paperwork, or worse, to follow, however! There have been a few cases of pilots landing when the RVR was below airfield minima, and all the ATCO could do was advise them of that fact and request their intentions, followed by: "There is no known traffic to affect your landing". The crews involved had a lot of explaining to do to the CAA!