Please read your pm again.. You currently have several posts of this nature but made no reference as to which one you were referring to in your pm or even whether they were linked in any way.
I am sure any reputable solicitor would establish their identities. My friend was a beneficiary in similar circumstances & the amount of proof she had to provide was incredible. Her birth cert, marriage cert, parents certs etc etc
Have you tried 192.com or the online phone directory? As you know the children's names - have you looked for marriages etc or used social media to try & trace them?