When our son, Isaac, was born in the Philippines we registered him and applied for a British passport. Once we got his birth certificate and our marriage certificate from the NSO (I think it took 3 months) and all other documents required, we registered him at the British Embassy and he now has a British style birth certificate and passport, therefore, in my opinion he is British and has the right to settle here.
When they came to live with me this year, my wife entered with her spouse visa and my son with his British passport.
Your wife should be entitled to all NHS treatment as soon as she arrives here. On a cost only basis it would be better to have your baby here.