Seconding shouldwestay - I was doing the school run, as a lone parent, with a 2 day old baby and a Yr 4 child. My DD felt massively sidelined, because - in her mind - I was no longer doing whatever she thought I was doing, but spending the entire day cooing at and paying attention to her new brother (in actual fact, I was frantically finishing a degree dissertation and panicking about how I was going to cope long-term as a lone parent to two). I had no support whatsoever, so, like many PP's, I coped. Because I had to. Because my older child, not to mention my then newborn, needed me to.
However, having said that... when my daughter was 4 and given a place at a school who thought it perfectly acceptable for their pupils to escape on a regular basis right next to a main road... I kept her out for the extra year and found her a far better school where she would be safe.
Only you know what is best, longterm, for your family, OP, but your son having been given a place at a school 10 minutes away isn't the end of the world. If you have support... get someone else to take him whilst you're recovering from the 'C'-section (please don't drive until the 6 weeks are up, and you're cleared by a doctor, because you won't be insured... as my SIL found out, when she was 15 days post-partum with a 'section and doing the school run and someone rear-ended her). If you don't... appeal and cite this as the reason why. You might be lucky. But if you're not... then he doesn't need to start school, legally, until he's 5 - as PP's have said - and he might relish/need the time with you and his new sibling in order to bond. I didn't get that luxury with my two, because my oldest was too entrenched in the education system by then. But if it had been an option? She would have been at home, with us, bonding with her baby brother and realising that I wasn't faffing about doing nothing, but working my socks off to provide for them both.
