People who cannot tolerate cow's milk dairy can sometimes tolerate other forms of milk, especially goat. A friend of mine has found her very sensitive (skin problems) daughter can cope with goat products (milk, butter, cheese and yoghurt). There is also sheeps milk, yoghurt and cheese.
I also understand there is a lactose free milk available that many people tolerate and which is sold in many supermarkets.
You may have to shop around for these, but Waitrose often has all the above.
You might also want to explore soy products. You can get various different types of soya milk and cream, but as a diabetic you'd need to make sure it is a no added sugar product. The differnt manufacturers products taste a little differnt, so it may be worth experimenting with different brands. You can get small cartons to try in some health/whole food shops.
Earlier this year I had a brief dalliance with dairy free, as we were considering going dairy free at the suggestion of a doctor who saw my (non-vegetarian) dyspraxic son. He suggested that it may benefit his concentration abilities by being dairy free for a while, before reintroducing foods to see their effect. In the end we didn't do this because of my diabetes diagnosis.
There are other non-dairy "milk" products on the market. I particualrly liked the oat milk and rice milk. Unfortunatly, both of these products sent my blood sugar through the roof :| There is also an almond milk that I have not yet tried, but plan to have ago with soon.
I had this for a (very) late breakfast this morning - out of one of Rose Elliot's low carb veggie books, she calls it Fabulous Frapuccino. I teasp of decent quality instant coffee, 1 scoop of soya protien isolate powder, 150ml water (although I replaced the water with soya milk), 55g tofu, 2 tablesp soya cream, sweetener of your choice to taste. Whizz it all up together until smooth and creamy, drink it staight away. Only 1.7 carbs if you use the water and just over 2 if you use soya milk, high protien tho...
It was really nice.