It is perfectly feasible for a limited company to go bust while a parent company is healthy (or to be sold then go bust - SFT had been owned by some very big backers a few years ago). It has happened on many occasions in other industries, though I would say not as far as I know in aviation training. I would also add that Oxfords assets are rumoured to include the airport and have very high value, in which case liquidation is likely to be orderly and lead to creditors being paid.
Do you have any reason to say BBA will not go bust? I have no idea how sound the company is, though I know it is large. It is smaller than Enron.
I think OATS going bust is extremely unlikely, but that does not mean people should not take great care of quantities of money most could not possibly replace, and ask questions. OATS should be willing to answer them, in the same way my bosses do when the same questions are asked of us. All schools are asked these questions. If they are of no help, then be wary.