As stated before, you wont be evicted on the 5th September, that is just the earliest date the LL can apply to court to make a claim for possession.
If possession order is granted - that will have 2 weeks before expiry.
Once the PO expires- the LL then has to apply to court for a bailiffs warrant.
You can ask for 42 days extension if you haven't anywhere to live
The council only have a "prevention duty" to you at the moment- not a duty to rehouse.
If a PO is granted then they will owe you relief and interim duty.
Under relief they may well help you with upfront moving costs such as a deposit.
They can also discharge this duty by offering Private rent as long as it is available for at least 6 months and is suitable.
Interim duty means they will have to house you in emergency accommodation as you are considered priority need as you have children. if they go on to owe you the main housing duty, (am sure your homeless officer has explained this) then they can discharge the duty by offering you private rent as long as it is suitable and available for occupation for a minimum of 12 months- or a social tenancy.
Depending on where you are - you could be in temp accommodation for a while.
S21 may be invalid- depending on the year your tenancy started/renewed- LL as again stated before has to have followed certain obligations with regard to Deposit, gas cert, EPC, How to rent - if it is invalid, LL will have to put all of these things right and then serve another s21 - delaying the process.
If you receive any papers from the court and the s21 is invalid - you will need to complete the defense form and return to the court.
This will then stop a PO being granted.
There could be hearing or if LL uses accelerated possession proceedings then there wont be a hearing but you will still need to complete the defence form if there is a defence otherwise PO will be automatically granted.
Worst case- S21 is valid - in ordinary times eviction takes around 7 months - this is going to be a lot longer with back logs in the courts.
You do not have to leave on 5th September.
If you are already in receipt of housing costs of UC (or Housing benefit if you are on legacy benefits) then you can apply for DHP for rent in advance./deposit but you will still need to find a property first.
You can apply through your council's website.