You need to get him back in the buggy. Start by using the buggy when it is not the school run, and give lots of positive associations, go somewhere he likes when he gets in, snacks, toys. Never use it as a restraining device/make negative associations. If necessary buy a new buggy, which is large and comfortable. You have two more years of this and you will not regret your investment!!
It is the transition/change of environment that will be bothering him, that ,and the fact that it is a stressful time of day, and he will be picking up on the fact that it is a demand. After school he can then do some running around in a safe place if he wants. All the people and the noise of the school environment will also be a trigger for him, possibly once he is there he realises it is not so bad, but like all anxiety it builds in advance.
The other good thing about a buggy is that you can avoid the whole getting the coat on boots on bit, he could actually go in his pjs if you have a snuggly buggy blanket/sleeping bag thing, and then go home and get dressed after the school run.
You could also do a social story about going to school, ie a talk through in advance about getting in the buggy, getting ready (if he likes putting his coat and shoes on) having his breakfast and coming home again afterwards and playing (sorry wrong order, obv breakfast first). First this, then this, then this, with little pictures. You might find some pictures books with stories about getting ready for school.
Luckily I had to drop my older one off first, and we were right next to school, but i can remember it took forever to walk to nursery and there was no hurrying them, lots of detours. However, they arrived happy and exercised if a little late..
I decided that it was better for them to be late for nursery than to have the meltdowns.