Imagine the challenge this presents for ATC. Departing aircraft could lift off at any point in the circle, and thus initially climb out on any compass heading, before (presumably) turning onto some form of "join up" heading until they reach the standard departure route.
So, there will have to be some form of circular "clear zone" for 360 degrees around the runway whenever an aircraft departs, and until it is safely established on some agreed heading. Then factor in the possible wind-affected "drift/fall" of vortices from departing heavies, which could blow back across part of the runway for a period afterwards.
If a single "starting point" is used for takeoffs, then departing aircraft will lift off on any heading, rather than into the prevailing wind. Unless, of course, they have some smart software to predict take off run, and thus to work back from the predicted "into wind" lift off point to an appropriate starting point, so that 'Rotate' happens just as the aircraft approaches an "into wind" heading.... So, ATC either has to accept aircraft leaving the runway on an unpredictable heading, or every single flight wanting to start it's t/o roll from a different point on the runway depending on aircraft type, windspeed/direction, and predicted t/o weight.
On a single runway airport, ATC will also have to slot arrivals (presumably coming in to land on a predictable "into wind arrival" heading) into the departing "spaghetti" of take-off flight paths and vortex allowances, whilst also allowing suitable 'go-around' clearance should the arriving flight need to abandon its' landing for any reason. Hmmm...
Also, and the radius of the circular runway will affect this, flightcrew will have to deal with a wind direction that continually varies throughout a takeoff, or landing, run - so a take off run may start with a crosswind from the left, that becomes a tailwind, that becomes a crosswind from the right, before finally turning into a headwind etc... And imagine the fun that crews will have if they take off with a significant crosswind element at point of lift-off, and the crosswind catches the raised wing as the still banked aircraft clears the "lip" of the bowl in which the banked runway sits

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Somehow, I suspect the new London runway at LHR, (or LGW

) will be traditional long straight strip of tarmac/concrete!
I guess the advantages of a circular runway are that landing approaches could always be made perfectly into wind, as touchdowns could be made on any compass heading to suit the wind direction at the time, and possibly, initial departure climbs might also occur at any point in the compass, thus spreading the noise nuisance, and post take off accident risk, over a wider area, rather than over the same patches of land at each end of a straight runway

.