Oats are a really good idea actually.
As are the little bottles of soy milk. Not sure if you have them where you are, but I usually pick up a six-pack of them before I go on any trips. That way, I can order a coffee, ask them to not fill it up completely, then slip the milk in once it's done.
In new work situations, it's tricky. If you really don't feel comfortable with letting someone know you're vegan, then don't let them. It took me a while before I was ready to announce my vegetarianism (and most people still don't know I don't eat eggs or dairy). I think feeling good about your own changes, feeling strong in your convictions, takes time. I don't think there's anything wrong with taking that time.
So, this is what I would do. Whether you bring food or not, if someone asks what you're eating, just tell them! After all, food is food. If you're good at baking, then one day you might want to bring muffins or biscuits for everyone to try. You can say they're 'egg and dairy free, for anyone with an intolerance'. Not only does that mean you have something to eat, it's a good way to start conversations with people.
And no one I ever know, has turned down a choc-chip muffin. :P