Update - British Airways (along with many other OneWorld alliance airlines) have recently
enhanced changed their policy on through checking. BA policy is now not to through check unless on the same ticket or booking reference number:
From 1 June 2016, the British Airways policy on accepting customers
travelling on separate tickets changed. Baggage will normally be
checked through to your final destination, provided your journey is on
the same ticket or booking reference. However, if you hold separate
tickets for the flights in your journey, we will only check your
baggage to your final destination if the connecting flights are
contained within the same booking reference, for flights operated by
British Airways or another oneworld airline.
If you have to recheck baggage for subsequent connections bear in mind you may need to go landside and clear border controls - that may also mean getting a travel visa. As stated previously in this answer speak with the airline responsible for the first leg of your journey.
For anyone else looking at this question from Google the quick answer is - it depends. Speak with the airline responsible for the first leg of your journey. BA policy is:
Transfer passengers holding through tickets - British Airways will effect through-checkin (including the through-checking of bags)
where applicable via the transfer point and minimum connecting times
will apply. In addition, in the case of delay or disruption at the
transfer point British Airways will apply involuntary rerouting
procedures which may include rebooking on to alternative flights or in
some circumstances hotel accommodation.
Transfer passengers holding separate tickets - British Airways will not effect any through check-in for passengers holding separate
tickets for onward travel from the transfer point, unless the fights
are BA to BA, or BA to oneWorld partner. Passengers who have separate
tickets will have to collect their baggage at the transfer point &
possibly clear customs and then check in again for their onward
flight. Extra time for the transfer will therefore have to be allowed
as minimum connecting times will not apply. Also, in the event of
delay or disruption at transfer point, British Airways will not be
responsible for any rerouting process because passengers hold separate
contracts for each separate journey.
Whilst they state they won't be responsible for rerouting for those holding separate tickets for each leg, I expect they would exercise some discretion for BA to BA transfers where reasonable connection times have been planned.