You will need to claim Child Benefit (CHB) in yir own name and if you go here,
http://www.hmrc.gov.uk/childbenefit/how-to-cla im.htm you can download the claim form. It's the HMRC that will process the claim and don't worry if you don't have the Birth Certificate, they'll have seen that when yir ex claimed.
They will be in touch wi yir ex to see if she wants to relinquish the CHB in favour of you. If she won't, the HMRC will decide who has the priority and let you know if you've been successful..