Section 21 housing - illegal squatters? England

Looking for some advice. I did post this on a housing Reddit, they suggested I post it here.

My aunt is living in a rented home with her adult son. The tenancy was is in her ex-partners name, but the landlords knew she and her son were there also. They have all been there around 9 years.

She is in good terms with her ex-partner. Landlord has given notice, aunt has been to council who have said don’t leave until after section 21 and then landlord has been through court process as they cannot afford to rent anywhere else despite searching.

Her ex-partner has let the landlord know he is no longer living there. Landlord has said aunt and her son are then illegal squatters and need to leave at end of section 21 notice or the ex-partner will incur all of the costs relating to court.

Is this correct, she wants to follow the council housings advice but she doesn’t want her ex-partner to be liable for any costs.

What is her best course of action other than speaking to housing again, who often don’t seem to know themselves.

Thank you in advance