What is the ruling on me lying to my mother and telling her that my brother kicked her out of the house, so that I can get a house in London because the authorities are not helping me obtain one?

The answer includes two matters:

First: Buying a house through a loan with interest is usury itself, and repentance for it is obligatory. If it is possible to get out of the usurious contract or to return the amount without interest, then do so. But if you are forced to pay the interest, then pay it, and the sin is upon them.

Second: Lying to obtain asylum or other services is forbidden and a betrayal of the covenant of security by which you entered the country. Therefore, fear Allah and do not lie or betray.