weaning a toddler off the bottle

Just seen this new section so have posted here as well as Toddler section!
I am currently trying to wean DD off her bottle. She doesn’t have one at night any more, but she has always had one first thing in the morning and then probably 3 or 4 throughout the day (one of the reasons I want to get her off them is that I think she is filling up on milk and not eating enough at mealtimes).

We went cold-turkey this morning which made her very upset (she usually gets up and has a bottle of milk on the sofa before getting dressed – she has breakfast when she gets to daycare). I can understand where she was coming from as I don’t like having my morning routine altered either! I offered her different sippy cups but she wasn’t having any of it. I managed to distract her in the end with getting dressed and ‘helping’ me in the shower, but she didn’t drink anything at all.

She drinks water out of sippy cups and normal cups at meal times with no bother at all so I know she can do it. And I know that she drinks milk out of sippy cups at daycare. (I have asked them not to give her any bottles today).

Anyways, I am looking for any tips that may make this easier. Is cold turkey the best approach? I feel like the meanest mum in the world when I think of her teary face this morning.