The whole movie is practically made of heartwarming moments, but for a small and subtle one: Daniel is Sam's stepfather, and all through the movie Sam has called him Daniel... right up until Daniel says they have to follow Joanna to the airport, to which Sam says, "Okay, Dad." He then immediately follows it up with a Funny Moment by adding, "Let's go get the shit kicked out of us by love."
Less subtly, Daniel's crush on Claudia Schiffer is a Running Gag in the movie (and played into a Refuge in AudacityFunny Moment in a cut scene). Shortly after the scene where Sam calls him Dad, Daniel bumps into a Claudia Schiffer lookalike (played by herself, naturally). The attraction is clear, mutual, and leaves him stammering... and he still focuses on getting Sam to the airport instead. And then, in the month later epilogue she's at the airport with them when Joanna gets back.
Juliet pops over to Mark's to pick up the wedding tape Mark made, which Mark desperately tries to claim he's misplaced and taped over, provokes Juliet to call him out on never liking her. As she watches the tape, she notes that not only is Mark making her look good, he has no moments where she's not on camera. She gradually realizes, to her shock, that the reason he's distant to her isn't because he hates her, but because he's in love with his best friend's wife.
Probably more of a Tear Jerker than heartwarming, but the scene where Karen confronts her husband for giving a gold necklace to another woman who he may or may not be having an affair with.
"Would you stay, knowing life would always be a little bit worse? Or would you cut and run?"
A rather subtle one when the Prime Minister sneaks into the Christmas concert and runs into his sister (Karen, as listed above, who has just found out about the necklace thing). She's shocked to see him and gives him this huge bear hug, visibly on the verge of tears, thinking he's there to see her kids perform- and clearly needing someone to lean on.
Jamie's proposal at the end, especially with a whole group of people following him to the restaurant.
Daisy's school's act in the talent show. It's ridiculously cute.
After Billy Mack makes #1, he's invited to a party at Elton John's, and we see his loyal manager sitting alone at Christmas. Then there's a knock at the door...and it turns out to be Billy, who's decided to abandon the party in order to spend Christmas with his one die-hard fan.