Landlord will legally have to give you a section 21 notice to ask you to leave, minimum of 2 months. Your agent should confirm this. Go to the local citizens advice centre to get some advice, most have appointments or drop in times.
A council may not look at housing options with you unless you have actually been given notice to leave the property or there is another reason why you cannot remain. You should be able to join the housing register (waiting list for social housing) I would do that ASAP. They should give you basic advice over the phone. Depending on where you live in the country some areas have waiting lists years long whilst others may be much quicker, all councils prioritise people on their own criteria for allocations to properties. It can vary a lot.
If you do have to leave the property the council will try to prevent you from becoming homeless by working with you to secure another property. If that is possible then it is unlikely social services would be involved unless there are other concerns for your Childrens welfare.
As others have said if you are found intentionally homeless (done or failed to do something that has led to the loss of the accommodation) or if they offer a property to prevent you from becoming homeless which you turn down and you have nowhere else to go, they may have a duty to make social services aware of your circumstances as you have children. Social services may contact you to check your housing plans. If you really have nowhere to stay and the council have no duty to help, they may assess you or consider helping or they may offer help only for your children.
I would try to get some confirmation from your agent about your landlords plans, if looking like you will be asked to leave start getting advice and making plans. Be proactive and try to take control of the situation. Far better than just waiting for the council to come up with something, as may be the last thing you want to consider.
Sorry for massive post!
Good luck.