If a landlord wanted to increase the rent, and the tenant give notice of leaving at that point, can they avoid the increase in rent during this two month period?
Assuming they are out of contract then in theory yes but in practice probably no. You are required to give at least a months notice of a rent increase. If I was you I would not bother for the sake of one month. Let them leave hassle free otherwise an upset tenant in your house in their last month could cost you lot more!