Private Landlords are always recommended as they don't do credit checks. Letting Agencies may be ok but they would want a higher deposit i.e. 6 months or so.
As a one time estate and letting agent, I would say that most agents do take credit checks (we used to in 99 out of 100 cases).
If an applicant said they had trouble from the start - I would waive the credit check (no point in wasting time effort etc) I was probably an exception rather than the rule.
Private landlords not likely to do credit checks - you will need a month in advance and a months deposit.
Try the local council to see if they will assist you - if you have children or a health issue
I am in the process of moving and have been up front with my letting agent. I have shown them my BR and discharge paperwork and they are happy with that. They didn't see it as a block to my being a tenant.