My mother lives in a £250,000 house.

She owns a half of the property - £125,000K

My brother and I each own a quarter of the remaining £125,000k.

When she deceases we will not pay inheritance tax because it is under the £300,000 nil band.

But will we have any capital gains tax to pay in the first instance.

Moving further would we have capital gains tax to pay if we sell the property afterwards.

If so at what rate.

Is it a avoidable to pay the CGT in any way?

For what country is this? Tax laws vary.

If it were in the U.S. you wouldn't pay capital gains tax because it is not income to you; it is an inheritance (subject to the inheritance tax). But since you are in the U.K. evidently, I don't know. As ScottGem said, "Tax laws vary."