Hi there
You should approach the council again and make a homeless application. You can also apply to go on the waiting list a (2 separate processes) but a homeless application can give you higher banding on the waiting list.
I'm not too sure why they are asking you for a letter at this stage as the onus is on the council to prove that you are not homeless, not for you to prove that you are. Anyway if it helps your case it will be worth taking it in with you so it should be addressed to you.
As you are an "excluded occupier" (unless you are married) then only reasonable notice is required. What is reasonable can be considered as little as 24 hours.
Anyway, right now -you are threatened with homelessness and eligible - the council now have a duty to help prevent you becoming homeless, which could be helping you to find alternative accommodation. (new law so not sure how councils are doing this at present)but in truth I can't see them doing a great deal- might be as little as giving you a list of private landlords etc)
However, once you are actually homeless, the council then have a duty to relieve your homelessness - again this means helping you find alternative accommodation (not necessarily housing you)but it could mean that they find you accommodation in the private rented sector as long as it is available for at least 6 months.
You will be considered priority need as you are expecting, which means they must house you under the interim duty.
This will mean them providing you with emergency, short term accommodation whilst they look into two more things - intentionality and local connection.
It doesn't appear from what you have said that you have made yourself intentionally homeless as you have been asked to leave and I am presuming that you have lived in the area at least 6 out of the last 12 months.
If you "pass" these last 2 steps then they will have a main duty to house you and they must give you a decision letter (s184) as to what duty they owe. (letter should be usually within 33 days)
If they decide that they do owe you the main housing duty, you may then be moved into temporary accommodation until they can discharge the main duty. This can be private rented accommodation or social housing from the waiting list.
I hope this information helps.
Good luck.