There's a surprising amount of cereals, both branded and supermarket own, that are vegan friendly. They're my go-to day to day.
I also like to make overnight oats using rolled oats, soya yogurt, fruit (fresh, dried and frozen) and spices like cinnamon and nutmeg. That's a good one to put in a tub and take with you on the go because it's quite thick and can't spill, especially in a decent pot.
Then there's good old chocolate on toast! My favourite is this one from Pip & Nut 
I've never had to buy breakfast on the run yet but can imagine the choice in cafe's is quite limited. Everything has dairy in it, let alone eggs and honey. I try to make sure that if I'm short on time I eat breakfast and sacrifice something else I don't have time for instead, like putting on make up. But that's just my personal choice 