Of course they won't! They will pass on the increases to the consumer! The only way the airlines will go bust, is if we all stop using them!! It is not 'rip off Britain' it is a standard worldwide business practise - unfortuantely it is always the end user (consumer) that has to pay for it! Some of the larger, and smaller, airlines may stop flying under subscribed routes and some airlines may amalgamate or be taken over by another, as a way of staying in business, but they will not go bust as long as there is fuel available and sufficient people that want (or need) to fly.
Some of the small ones already have. Max Jet which used to operate a business class plane service in London went under and closed its door. Some of the other small independents will probably follow suit unfortunately which will reduce the competition and will see prices rise. The large companies will just increase their taxes and charges so that their tickets appear the same price till you get to the final piece where you actually have to pay then you will get a nasty surprise that your from £19 air ticket will cost your £75 or so. Happening everywhere and happens naturally with the increase in price of living just that with the oil costs over here it is going to jump further than we really want it to.
It depends on whether they have realistic prices for the flights. Flights by low cost carriers have been cheaper in the last few years than they have ever been. £1 flights are not sustainable. It's perfectly reasonable to pay £100 return to fly to Spain/France/Italy. If they start pricing at this amount and stop the £1 bargains and the expensive last minute price rises the airlines have a good chance of continuing their operations. They will also need to cut the less popular routes.
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