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I ride a UA for work (not the family though) and used to ride a Bullitt. Both ride totally fine once you're up and running, the UA position is more upright, at least with the stock stem/bar combo. The only thing to consider is the slightly larger size of the UA, it makes it a bit more difficult to manoeuvre at low speed and gives you a bigger turning circle. One upside to the UA is the step through frame, I find it a lot easier to get a heavy cargo bike on to it's stand when I'm not on it and it feels a lot easier to control the weight of the bike as you're getting off with the Urban Arrow. Honestly though, it won't take you long to get confident or comfortable riding either of them, they're not that drastically different.
How different is the handling of a UA family and a bullitt?