I hold both a Bahraini and a British passport. There is a slight difference ( one letter ) n my first name. To travel from Bahrain to the UK can I show the BA check in desk my Bahraini passport. It has no UK visa in it. Is it appropriate to show the British passport? The Advanced passenger information system that is requested by BA where does it go? To the country entered or that exited or both?
As a rule, at airline check-in, you need to show the airline the documents that prove that you can transit any countries that you may be transiting, and that you can enter the destination country.
If you travel on a direct flight from Bahrain to the UK, then you need to show the airline the documents that prove you can enter the UK. In your case, this is the British passport.
Note that this is different to what you need to show to exit immigration! You should show them the same passport you entered the country with (most likely your Bahrain passport).
The APIS information is sent to the destination country (UK).
And typically a small spelling error in your name is either ignored, or the airline can easily change the name on your booking. But you will want to have the passport with the spelling error reissued as soon as you can do.
In order to board a BA flight to the UK with your Bahraini passport, it needs to have an entry clearance in it. Since you don't have one, the answer is you should use your British passport. You should make the reservations with that passport so that your name will sync up to your travel document. Michael Hampton's answer addresses how to handle Bahraini exit controls.
For your second question about where the information goes, it goes to the Border Force whenever you enter or exit the UK on a registered passenger carrier (or more to the point - whenever they ask for it). The information you give gets combined with other information like your credit card details and the number of pieces of baggage you checked and the list goes on and on.
It's available to the Border Force before you arrive and after you leave. If they think it's interesting there are agreements with other countries to let them know. I don't know the case for Bahrain, but it's likely they follow the same model.
The UK's solution for dual nationals who want to conceal their British nationality has been to issue the person a Right of Abode certificate. This certificate does not mention nationality at all but allows the person to travel to the UK without a British passport. You may want to see if you qualify for a Right of Abode certificate. On a personal note... They are not easy to get. For example I do not qualify based upon the nationalities in my other passports along with an unexpired British passport.