I just got back from Ibiza with baby without dad and wasn’t asked any questions there or in the UK, and they were doing extensive checks in UK. I didn’t bring a letter or anything didn’t even think about it tbh! But all was fine
That’s flying from Gatwick, it probably helps that me and my son look so similar
I flew home to visit family with baby and without dad. Our son has both our surnames. When I arrived, immigration asked a couple of questions and suggested a letter would be a good idea for any future travels.
I have flown solo a couple times with my daughter who has my husbands surname. I’ve only been asked for anything when arriving back in the UK and there are signs up at Heathrow letting families know they may be questioned. All you need to show is a birth certificate and you’ll be fine. I also had signed letter, but was never asked for it