Is the relative who owns the flat living there with you too?
No only myself.
The council and the court will treat you as a tenant even though you have no formal lease or tenancy agreement.As such in order to make you leave they would have to give you a formal notice to quit containing a date upon which you should leave. If you fail to leave on that date then they have to go to court to obtain a court order which will order you to leave by a certain date.If you fail to leave by the deadline they will then apply for the bailiffs to come to the property and take possession which they do by changing the locks.You are at the stage where notice to quit has been given but the deadline for the Notice expiring has been extended so you are pretty much at the start of the procedure.
Being exact on how long it takes is hard because it depends on how busy the courts are for fitting in the applications to court of the council , how efficient the council are in taking each step and the availability of the bailiffs but on average it takes about 2 months.
The council will not let you stay because they have a list of people waiting for housing so you would have to go to the council and apply for housing.Your priority need is determined on a points basis.
I hope this clarifies the position. If there are any further points please reply.