It does not matter that you do not have a written agreement in place, the tenancy will be held to be a assured shorthold tenancy.
Under Section 8 Ground 8 of Housing Act 1988 a Landlord is entitled to take possession of the property is the Tenant is at least two months in arrears with their rent. You must serve a s.8 Notice on them giving them two weeks to vacate and you can download the relevant form from www.lawdepot.com
If they do not move out of the property after the two week notice period has expired you can make a claim for possession of the property through www.possessionclaim.gov.uk/pcol
If you do this you can also make a claim for the arrears in rent.
If they do vacate in the two week notice period then you can make a claim for the arrears by using www.moneyclaim.gov.uk but the fact that you have not got a written agreement will make it more difficult than it otherwise would be.
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If the tenants are at least two months in arrears with their rent then you can serve a Section 8 Ground Notice on them under the Housing Act
Sorry, I think I misread that part of the question. As there is nothing in writing, the Landlord is required to give the lodger "reasonable notice". "Reasonable" depends to a certain extent on whether the lodger pays rent weekly or monthly.
A months notice is appropriate in most cases in which rent is paid monthly, but if there is a child involved then he may consider giving a slightly longer notice period event though he obviously not under a duty to house a child who is of no relation to him.
The notice should be given to the lodger in writing, it's not essential to give reasons but if the lodger is not paying rent then this should be included as a reason.
If the lodger does not leave after the expiry of the notice period then the safest way for him to evict the lodger is to apply for a court order for eviction.
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