I was looking into this today. On the BMW.co.uk website it states:
* Your tyre(s) must have 6mm or more of tread depth at time of claim to receive 100% of costs. For part worn tyres with less than 6mm but more than or equal to 2mm, 50% of costs will be paid.
I also looked on the ala.co.uk website and that doesn't seem to list any such restriction for their tyre insurance so it would seem to be the much better policy to go for.