Owning the cassette does not give you legal rights to convert the audio to other formats or duplicate it beyond what is required for playback (temporary cached copies etc.) Thus in the UK even ripping CDs to your phone/MP3 player is technically violating copyright.
However, copyright infringement is not a crime, so it isn't actually illegal unless done on a commercial scale for profit. At most, the copyright holder could sue you for damages, in this case the price of the replacement MP3, typically around 70p. In practice copyright holders generally don't bother with people making copies for the car or work, taping songs from the radio, time-shifting, ripping to their computer and the like, but there is no guarantee that they wouldn't pursue you for that very small sum of money.
In addition, you must also consider how you would download the MP3. Peer-to-peer file sharing systems like BitTorrent also upload data as you download it, so you could also be sued for losses stemming from distribution if caught.