1. Dear Gissela, you would be best advised to have the birth in the UK. This is because the child automatically gets British citizenship by descent if your child is born in the UK. If the child is born in Ecuador, then you will have to have spent three years in the UK as well as register the birth before the child is 18 in order that the child gain British citizenship.
2. This three year period of residence can either be the three year period before the birth or the three year period before the registration of the birth is made. So you need to be able to live in the UK if you wish to get your child citizenship, whether before the birth for three years or for a three year period before the child is 18.
3. However, you can apply for UK citizenship by descent when only you, the mother, has British citizenship by descent, and the father does not hold UK citizenship at all. Citizenship now passes via the female line as well as through the male line.
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5. Three years up to 2011 does not count. They need to be the three years before the birth. Secondly, you will not get your British citizenship application refused if the child is born in the UK. Alternatively, you need to spend three years in the UK before the birth or before the child is 18.
6. You don't need to show you intend to live in the UK, for your child to get UK citizenship. That is not correct. the only rules are the ones I have outlined here.
7. The passport office has no role in the registration of the birth in the UK where your child is born in the UK. Anyway, it makes no difference if the father is present or not. You can still register his name on the birth certificate. Secondly, you are confusing what British citizenship by descent is. It makes no difference where you are born. If you are born to a British citizen parent who has held citizenship since birth, then you obtain citizenship by descent.
8. Here is a link to the addresses to which you send your application form if your child is born abroad. The address varies depending on where you are. https://www.gov.uk/register-british-citizen/where-to-send-your-application. The fee is GBP 936 for each child.