You'll be fine to take a backpack. You can take 2 pieces of cabin baggage on BA, their allowance is overly generous in my view!
1 handbag/laptop bag (max. 23kg / 51lb and up to 40 x 30 x 15cm / 16 x 12 x 6in)
plus
1 additional cabin bag (max. 23kg / 51lb and up to 56 x 45 x 25cm / 22 x 18 x 10in)