You will need to submit an
Out of Time (form PE2) application with your Statutory Declaration.
The OOT will also need to be witnessed. SDs and OOTs can be witnessed free of charge at any convenient County Court. Phone and check opening hours and make an appointment if necessary. Solicitors now charge £8-£10 per document to witness.
Before you submit please post up what you propose to put on the OOT application. This needs to be the reasons you are late and not anything to do with the original contravention.
For the purposes of these forms you are the respondent and TFL are the applicant.
When did you put the private plate on the car? Before or after the date of contravention?
Just because you received a ULEZ PCN, doesn't mean the car isn't compliant. There are errors and omissions in the database. One way it goes wrong is emissions data not transferring when a private plate is added.
Do you have any documents such as a Notice of Enforcement from the bailiffs? If so, please post it up. Just redact your name and address. Leave everything else visible including any reference numbers and the vehicle reg.