I suspect that there isn't a market to opened up.
Even if there are 150,000 UMs per year with BA, you would have to cover all the major routes and pay for the seat and the time of an experienced and competent child minder. Approvals and vetting would be a nightmare - especially across multiple countries.
Whilst one might be able to approach this rather like the old fashioned courier business, I think that trying to field staff across a wide network - when you only want them to work piece meal - would be difficult. Would you get agreement from families that you can look after children from multiple families at the same time, for a lower fee?
Two simple thoughts. Your company receives a request to escort two children on a flight and their parent's say - We've booked them in Club as usual. Will they pay another club fare?
Or, they are booked in Y but the flight is fully booked. They will not even accept an overbooking and early check-in, then you cannot get the business.
Long haul is another problem: Your escort is going to need to snooze at some point, whereas cabin crew have always got someone on lookout.
Many moons ago, BOAC had some 'aunty's that rode with the children who were travelling out/back to colonial homes/schools but that has gone. Now they are charging a supplement to look after them and it will work.
Whilst there might be room for a specialised agency, within a circle of friends and business associates, I doubt that it could work more widely. But I sit to be corrected.