In general, the minimum amount a player can deposit is £5 for the vast majority of bookmakers. This also covers several sign-up bonus offers, with the following all offering a deposit bonus off of an initial £5 deposit:
Ladbrokes (Bet £5 - Get £20)
Coral (Bet £5 - Get £20)
Betfair (Bet £5 - Get £20)
Tote (Bet £5 - Get £20)
Dazn Bet (Bet £5 - Get £5 + 10 free spins)