I've checked the government websites and cannot find any information or guidance about travelling with kids abroad when they have a different surname to you.
In 2008 we drove to France, via the tunnel, me, hubby and my eldest was 16. Peter stayed on in France for a few days and was flying back after some business and Jamie and I drove back.
We arrived at the port to be questioned about why we had booked for 3 but only 2 were returning, I explained to customs that the tunnel booking hadn't allowed us to book 3 out and 2 back and that my husband was flying back at a later date...OK no probs...but what is your relationship to the child in the vehicle...I said he was my son..why the different surnames...I explained I'd re married...did I have any proof he was my son...I had a scan photo in my wallet, just ask him, he's 16 FFS...eventually they let us through and suggested I carried his birth certificate with me when we travelled again....
...and here lies the problem...my son had my maiden name which was changed by deed pole when I married, then I got divorced and remarried...so to prove he was my I'd need to carry his birth certificate and mine to prove my maiden name, then my marriage certificate to show my previous name, then my new marriage certificate to prove my new name and make the connection back to my sons name...
So from then on I carried all this documentation with me when we flew anywhere...but was never asked for it, the US, Tunisia, Turkey.
When we moved to South Africa we had visas, we'd had to have a letter of permission from the children's father to obtain the visas...no problem leaving the UK, but when I tried to take the youngest out of the UK after a visit home last week, I went through the same problems.
FFS do these people not understand a Visa and the process I went through, no one anywhere tells you to carry all the above mentioned documents when travelling abroad with kids. They finally let us through in immigration at Heathrow, but advised I carried his birth certificate on the next visit...which still wouldn't prove he was my son, just that I had his birth certificate.