How long before calving does the udder start filling with milk?

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I would like a little more info before I answer. are you thinking you have a problem with a cow/heifer? what do you mean by "filling with milk?" They can start producing milk almost any time during third trimester but sometimes don't start filling with milk until after calving.
Never try to get milk out of a pregnant cow. The teats have a block/seal to prevent mastitis. You just opened a teat.
IF it was super easy to get milk, I would be worried. Either she has a calf sucking on her or she hasn't dried off from her previous calf.
Just because she doesn't have her own calf, doesn't mean another calf isn't sucking.