It does not matter that the tenant did not sign the tenancy agreement; it will be still held to be an contractual periodic tenancy and it will not effect your entitlement to take possession on the grounds of non-payment of rent. In future, to make things easier you should ensure that you obtain the tenants signature on the agreement.
Under Section 8 Ground 8 Housing Act 1988 a landlord is entitled to take possession of the property if the tenancy is two months in arrears with his rent (if the tenant usually pays monthly). If he is not in arrears for the full two months because of part payment then you could consider using Section 8 Ground 11 if the tenant has been persistently late in paying rent.
If you have served the notice on the tenant in the correct form, given him two weeks notice and that notice period has expired then you are entitled to take possession of the property.
If the Tenant does not vacate after the notice period then you should make a claim for possession through the following website:
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