I recently transitioned our 14 month old to whole milk, but I used goat's milk. She was used to having both pumped milk and also formula. I started by adding a little goat's milk to the formula, then more, and more every few days. (I imagine you could add a little to breast milk and gradually increase) In a couple of weeks it was all goat, and two months later she absolutely loves it and is self-weaning. She even begs for a bottle of milk when it's not even nap-time.
She had had a dairy allergy since birth, which is why she was on only breast milk and a hypoallergenic formula. The allergist tested and cleared her at 1 year to go ahead and try regular milk, but she had a bad reaction to cow's milk products that landed us at the pediatrician.
After a little break and back to the formula, I put her on goat's milk and she hasn't had a single problem. It's rich and creamy, has a smaller protein, so its vastly easier to digest, it has lower lactose. Goat's milk casein is more similar to human milk.
I got this from Dr. Sears: "Although the mineral content of goat's milk and cow's milk is generally similar, goat's milk contains 13 percent more calcium, 25 percent more vitamin B-6, 47 percent more vitamin A, 134 percent more potassium, and three times more niacin. It is also four times higher in copper. Goat's milk also contains 27 percent more of the antioxidant selenium than cow's milk." (It says though that cow's milk contains five times as much vitamin B-12 as goat's milk and ten times as much folic acid, so a child on goat's milk needs to be supplemented with those vitamins.)