Hi there Kiwifruit
I am a family lawyer based in Clapham Common. The answer to your question is, unfortunately, not black and white.
As one or two of the other replies on this thread indicates, if the father's name is on the birth certificate and your eldest was born after 2003, then he will have parental responsibility (PR) and you will require his permission to remove your eldest from the jurisdiction, or otherwise the court would have to decide the issue in what a "Leave to Remove" application.
If you left with your eldest and he has PR and objects then it would be an abduction and he can apply for the child's return - there is an international convention which applies in most western countries to deal with this.
Whether or not you would be successful depends on a whole host of factors, but the paramount principle is the child's welfare. If your eldest is a teenager, their ascertainable wishes and feelings are likely to be of central relevance to the question of what is in their best interests.
In recent years it has become more difficult for mothers in your position to obtain leave to remove (the case law used to be weighted in favour of the departing parent) and each case turns on its own facts, so they are notoriously difficult to call.
You may also be entitled to revert back to the father on his financial position, depending on his own circumstances. You can apply for capital and income provision from an unmarried parent via the courts as well as the Child Maintenance Service, but in order to advise on that I would need to know more about his circumstances and what he is paying at the moment to see whether you qualify.
If you want to chat it through, you can call me or drop me an email - my details can be found at
www.burgessmee.com
Good luck!
Peter Burgess