Just because there is no signed AST does not mean there is not one.
As such you can issue a S.21 notice:
Once notice has expired, which is currently 6 months
If you gave your tenant notice on or after 29 August 2020, the notice period must be at least 6 months.
Once that has expired you can apply for possession using form N5:
The Court will list the matter for hearing. The Judge will decide whether to give possession
Can I clarify anything for you about this today, please?
If you need anything further I am available for a follow up at no extra cost.