My 2DS are the same age and do this too but only when they're picked up in the car. If you're able to how about try walking home with them or parking away from the school to walk a little bit. The exercise and fresh air picks them up and if they don't want to talk to anyone they can run off and play. All being cooped up in a car is a recipe for disaster if everyone's in a bad mood, believe me the walk home from school keeps me sane! hope things improve for you.
Mine are like this sometimes. I have found that taking snacks not just in the car but right to the classroom door,and practically shoving them in their mouths as they come out, does the job. I'm not sure their hellishness is alwaya due to hunger, but the distraction of a snack, not to mention having something (other than shouting or crying) to occupy their mouths seems to work. Car is too late - World War 3 is often unstoppable by the time we get to it. I don't do this all the time - they seem to have brief phases of post-school horridness.
OP, could your DS1 be suffering from the onset of puberty? Lost of testosterone making him moody? It's not an excuse, it's an explanation. DD1 is now 12, she started this at 10-ish, and DD2 who is 10 is showing signs.
You need to start setting boundaries all over again as if they were toddlers in the throes of the terrible twos, it really does work (though obviously the language you use is different). There's some fasicinating neuroscience about what happens in a teenage brain, and it really is meltdown and back to the toddler stage. Calm voice, repetition, routine, boundaries and consequences - it's right back to the drawing board, but it's the way to go.
If it's any help, DD1 is much nicer at 12 than she was at 10 - despite teenage flouncing, angst and bitchiness, the mature adult she is going to be at 17-ish is already lurking in the background.
It's so stupid, because both teachers tell me how lovley it is that they hug whenever they see each other at school! They don't always argue together, iyswim, to be honest it's the eldest one's attitutde and temper which pisses us all off, he shouts at the younger one the younger one gets upset, I shout, we all shout, tra la la la laaaaa!
10 yo DS1 and 6 up DS2 both so moody and grumpy, bickering and yelling at each other 30secs after getting in car. yes I get they are tired and hungry and yes I have food in the car waiting, but FFS get a grip, it's hideous. I just flipped my lid with it, not the best response, but there you go. 10yo is the worst, he needs to get a grip with his temper and agressive attitutde. He won't stop despite being told 5 times!! Y6 is hard and all that and we always take it out on those we love, but I am now really pissed off with it, I may even go out to work and shove them in after schoool club so I don't have to be privvy to it. (I work from home now so fit in around them(