In reply to TClimb:
Go to Decathlon. If you don't know much about bikes, buying second hand may be tricky. Decathlon is probably the best value for money around and you should get something that suits.
Contenders are probably a Triban 500 Se or Flat bar road bikes - but they'll have thinner tyres.
Alternatively a Hoprider 520 City would do the job - that's £320.
the Triban 500 flat bar version comes in at £260, which gives you enough to buy a Decathlon child seat within your budget too (£40). You'd need to check that it would fit though. It would almost certainly fit on the Hoprider, but that's now bursting the budget somewhat.