Hi there; If the landlord doesn't want to offer a new tenancy, then there is not much you can do. Local Authorities frequently advise that you are not homeless untill actually thrown out by bailiffs. This is not true. Homeless person is someone
threatend with homelessness within 28 days, or has no access to available, reasonable to occupy accommodation. Each of these terms (available, reasonable o occupy) carry great weight. However, due to severe shortage of social housing Local Authorities will dig their heels and refuse to help until you are virtually on the street. So you need an advocate.
Shelter: Advice and support can provide you with local homelessness advice point. Your council should also have a homeless persons unit, which should offer other help; re-negotiation with the landlord, list of landlords who are prepared to take on HB tenants, rent deposit scheme.
As a person with young dependants (with presumably no restrictions on your immigration status) you are in a priority group.
Can you also tell us a little bit more about your tenancy and the notice: when was the notice issued and when does it expire? When did the tenancy start and when is it due to end? How was the notice entitled? Is it a private landlord, housing association or registered social landlord?
Call Shelter and don't be too disheartened; who knows, maybe you will find even better place to live?