She can run up a dress in a couple of hours on her sewing machine.
Do American English speakers use "run up" to mean "to make something very quickly, especially a piece of clothing, by sewing"?
I haven't heard it, but it sounds correct. It could be that it's a regional expression, not common in all parts of the US. It could also be an expression used in the clothing or fabric business.
I've heard of definitions number 1 and number 2. Definition number 3, the one you've asked about, is new to me. Number 4 sounds familiar, but I can't say I've said it or heard it often.
1. Make or become greater or larger, as in That offer will run up the price of the stock. [Late 1500s]
2. Accumulate, as in She ran up huge bills at the florist. [First half of 1700s]
3. Sew rapidly, as in I can run up some new curtains for the kitchen. [Mid-1800s]
4. Raise a flag, as in Let's run up the flag in time for the holiday. This usage, originating in the navy about 1900, gave rise to the slangy phrase, Let's run it up the flagpole and see if anybody salutes, meaning, "Let's try this out." The latter originated about 1960 as advertising jargon.
Is it rare in British English to use "run up" in the sense of "to make a piece of clothing quickly, especially by sewing"?
I don't know. To me, it sounds more American than British, but I can't say for sure whether or not they use it in British English. The expression originated in the mid-1800s. This might mean that it originated in the US, but I don't know for sure.