No, I do not believe that am and fm radio transmissions are likely to cease in the near or even medium term future, so keep your FM radio.
A lot of nonsense has been used to scare people into buying new DAB tuners. I suppose it's good for business, but not so good for your wallet!
Bear in mind if the BBC were to stop broadcasting on the FM bands, pirate radio stations would spring up in no time to use the freed up frequencies. Unlike other frequencies, the authorities would not be able to sell the FM band frequencies to private low power mobile users as the frequencies would be filled up very quickly by high powered pirates, so the government gains no benefit, (a pirate can easily run a transmitter delivering an ERP of half a kilowatt yet most private radio users use powers under 10 watts - some use fractions of a watt). Pirate radio is dangerous to the government as it can so easily carry political messages; I have heard it once and unlike most, found it rather worrying. Just imagine a pirate run by any kind of extremists - whether right wing, left wing, criminal or simply foreign, peddling illegal goods or a harmful political message.
Tracking pirates down is expensive and slow; the best method of stopping pirate radio stations is to keep the bands full by continuing the BBC and existing commercial radio transmissions. At the same time, construction of a powerful pirate radio station has never been easier; the transmitters are readily available from abroad at modest cost, and the rest of the station can be put together with parts from any DJ equipment shop! The only economically viable protection that I can see is continuing the present FM radio system to block all spare spaces on the FM band.
How do you track them - it can be almost impossible; a pirate radio transmitter with a ten to twenty mile range on a vacant frequency could even be fitted into a car, using what appears to be a normal car radio aerial! As nothing is visible and it moves, tracking is near impossible. Now imagine 50 such transmitters just in the London area!
I know it's not guaranteed, but I think, given the vast number of FM radios in the country, it is most unlikely that we'll see the FM bands go silent in our lifetimes.
Incidentally, I often find users of the new digital equipment wondering why reception is unsatisfactory. The reason is that the service is not magic; it is ordinary radio on what used to be frequencies used for band 3 television, (the original rather eratic band used for early ITV transmissions - do you remember the set top aerials that you had to move around for a satisfactory signal?). For stable and reliable reception if you do not happen to be close to the nearest transmitter you will need an outside aerial, i.e. the same rules apply as for FM radio.
I hope this has helped.
To the right is the author of the above article, Mike Solomons, owner of London Sound.
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