I have to strongly disagree with the other answers, because I think they're offering dangerous advice (although it may be a cultural thing).
My take on this is that it's the managers style, he's just given you a heads up without making a formal issue out of it (however harsh it seems, that's what it is. You could have had a written reprimand entered into your employee records instead, you got a break). It's up to you how to take the heads up, pushing back could easily lead to disciplinary action. A real swine of a manager wouldn't have given you a break.
In time or further in your career you may come to appreciate this no-nonsense management style. It's about focusing on the work during work, not the life dramas and it can have many benefits to all concerned. And similar managers I have known ran a tight, efficient team and were absolutely to be relied upon to accomplish things. Some cantankerous managers are very highly valued both by the company and their teams. Always attempt to see positives in people before dwelling on negatives because they inconvenience you.
This sort of flexibility is usually at the managers discretion for their own team, rather than company policy. New Manager, new discretion.
So listen to the manager. Managing your personal life is your problem. I know it can be difficult, I have 4 kids, but you just have to organise it properly.
The odd lateness due to unforeseen circumstances is usually OK, but more often is disruptive to some workplaces and in others you just have managers that don't care. You're not the only mother working, most handle the situation without their workplace needing to get involved at all.
My best advice is make sure that you don't have to rush, plan a healthy time margin somehow and don't rely too much on others to make it work, because other people have lives as well which can interfere with your plan.