Hurdity has given some great advice and info.
I doubt it is the oestrogen patches causing the nausea. I actually find my nausea is worse when not on oestrogen!!
I would definitely say that nausea and digestive problems are linked with the menopause - especially if they have emerged at the time of meno.
I get these episodes of nausea with pain around my eyes and forehead - sometimes for up to 3 days. I think they are linked to fatigue due to lack of sleep as they often occur when I've had couple of particularly bad nights.
I can't say that milk or lactose causes my nausea - I have tried stopping dairy with not benefit. In fact, when I have these nausea/headache episodes I find milk can help calm things down!!
As Hurdity has said, I also doubt progesterone is the answer - progesterones often trigger headaches and nausea.
My bouts of nausea have increased since I came off systemic HRT nearly 5 months ago and I'm trying a tablet the pharmacist has given me that you dissolve on your gum when the nausea hits - it's fro nausea associated with migraine. This leads me to ask whether you are perhaps getting silent migraines?
I agree with Hurdity that maybe the oestrogen dose might be worth looking at and I always liked using the Oestrogel as I found it gave a more consistent dose, as it is applied each day, and you can adjust the dosage as well - start really low and only increase if flushes aren't under control.
I don't think you told us what dose of patch you were using? Also, how old are you?
I would also have a good look at your diet - maybe try cutting out dairy to see if it helps but get advice from a professional over diet as you don't want to be low on calcium or other vital nutrients. Eating small amount often is also vital. Keep us posted DG x