It is exactly the same covenant that can be applied to any business, other than a doctor or solicitor. The covenant makes no mention of noise or traffic, in fact as has been pointed out, a doctor or dentist would certainly generate more journeys.
The planning authority were happy that the nursery met all conditions for such a business set up in a residential neighbourhood.
I know that children playing outside can generate a level of noise that you would hear if you had windows open. The planning permission was granted on condition that the hours spent in the garden are limited, certainly not before 10am or after 5pm, and with a break for lunch. This was considered a reasonable condition to protect neighbours from excessive noise.

I can confirm that, as a parent, I have been given notice that my daughter's nursery place will no longer be available in three weeks time. I do not understand the exact details of the court case in December or the land tribunal hearings, but I have little doubt that if the covenant is presented to a court, that the nursery will need to close immediately.

The new nursery in East Dulwich is lovely but it already has some children attending, so there will probably not be room for all children. Additional running costs also mean it will cost us more. It is a 30-40 minute walk for my wife each way, twice a day. It was much more convenient to walk up the road (10-15 minutes) to the street where my parents live. The Fairlawn nursery is for children over 2 only, and is not for such long hours.
My wife and I (and many other parents) have some very difficult life choices to make in the next couple of weeks. I can assure you that this is no less stressful than the sound of my daughter happily playing with her friends!

If action against the nursery is stopped then not only will it help me personally, but it will increase your house value. There are many parents who would happily pay above the odds for a beautiful detached house close to one of the best nurseries in the area.

I would like to think that it should be possible for the nursery to continue to operate and for neighbours to discuss their concerns with the owners (which has never happened), rather than using the 1937 covenant to terminate all childcare activity on the site. Perhaps there is a number of children on the site that would be acceptable to both sides - 8, 12, 16, whatever? I don't think the forum is a good place to negotiate over such details, but if there is a way that the nursery would be acceptable to you, I am sure that the owners would be interested in listening. If there is anything I can do to help neighbours work out their differences I would be happy to assist.