Firstly, Qatar is not in the UAE. Citizens of 100+ countries do not need visas to enter Qatar so that may not be an issue anyway.

Secondly, BA's policy is not to check through on separate tickets. Ordinarily the baggage policy of the ticket issuer would apply but you may be lucky at QR check in at SFO.

If not then I would suggest not to panic. Go to the transit desks on arrival at DOH. They will be able to check you in for the second flight and issue a boarding pass. The complication is if you have checked bags. If not then all is good. If yes then you will need to ask the transit desk people if they can arrange for your bag to be retagged to COK without you having to collect it and clear customs. On a longer transfer this should be fine but yours may be a little tight. All you can do is see what they say, or reschedule your onward flight to be sure.

All being well you will be fine. But be aware that the risk with separate tickets is always on the passenger. Even if they do check you through, your "connection" is not protected because it is not one ticket (and in this case is not even a QR ticket). A delay to the inbound could still mean you missing the COK flight and having to buy a new ticket.

Also be aware that because your ticket from SFO is only to DOH you may need to demonstrate at SFO that you are permitted to enter Qatar either visa free or with an issued visa, or they may refuse to let you board the first flight. So it is worth checking the visa situation for your nationality carefully before the flight. Alternatively, see if you can rebook your itinerary as one ticket.

As before, hopefully you'll be fine anyway but please be aware of the possible pitfalls of travelling on separate tickets.