What you don't say is how far it is to nursery. But it's clearly walkable.
I think for a one-off special occasion or a last-minute disaster, giving a lift would be kind - if you're available, which depends on your work. So the only reason I'm wavering here is that it was his first day at the new job, which could be considered special. But, he should have arranged his special lift with you in advance. (And what are the chances he'd have been late for that, kept you waiting and still have been late / only just on time?)
Medical explanations aside, there are plenty of people who behave the way he does because they're faffers, procrastinators, underestimate how long things take, strategically incompetent / resentful at having to do the nursery drop-off, or just poor at time-management.
He needs to adjust his ideas about time and recognise what his real 'leaving the house' deadline is, then work back from there, making sure he allocates realistic timings to each task. He needs timed milestones, not just the final deadline.
He may well be like many people I know, who sit around, relaxed and complacent, chatting and playing over breakfast time, feeling like they have all the time in the world, until the point at which they suddenly snap into action and go into a mad rush to get out the door - always too late, because they think they can do 20 minutes worth of getting ready in the last five minutes.
As for I will never get over the sheer selfishness that posters on here show towards their own families I would say that his expecting you to drop everything and rescue him, without any prior warning or arrangement, would be extremely selfish, yes.
The car thing is odd though. It's a family car that benefits all members of the family, so why is it not a family expense? Do you think he'd refuse to pay for it? Even though it benefits his wife and his child? That's your implication. Or that he'd start to demand that you chauffer him round like a taxi service? There's something really off there, about his and/or your ideas of family, sharing and caring.
If it was your own two-seater sports car, that didn't take a child seat and you used only for your own fun, fair enough. It isn't though.
Do you find yourself buying nice things for your child out of your own money, to avoid having to justify 'frivolous' expenditure to him? That's a really slippery slope towards you being the 'baby mama', him a single and relatively much better off man, who just happens to live with you. With the car, you're already part way down that slope.