I have traveled many times with my son alone to Spain as my family live there. I have only ever had it flagged up once, and the boarder control guy just said that some colleagues may question it , and it's advisable to keep a pic of birth certificate on my phone in future. Nothing was mentioned about permissions etc, so can't help with that, this is just my experience.
I second @Stephanie comment. We travelled last year, and we had a form signed by husband containing all our details, his contact details, where we were going, and for how long, and it was signed by a 3rd party as well. The form is from the gov website.
Legally you have to have permission to take any child out of the country, from anyone with parental responsibility for the child. That is technically the same requirement whether the child is yours and whether you have the same surname or not, but typically you are more likely to get asked if your surnames are different because they may also want to prove your own relationship to them. The advice is to take the birth certificate and it needs to match yours and their passports. If the dad is on the birth certificate they have automatically got parental responsibility and so you do need their permission also. There isn't a way to use your own name instead. The only way you can legally avoid it is if you have a court order (it's called a child arrangement order), which allows you to take them for up to 28 days. However, I know people who haven't been asked when theyve travelled and people who have, so depends if you're willing to risk it I guess.