The EU Settlement Scheme was designed for EU/EEA/Swiss citizens who came to the UK before January 2021 and wish to continue living here, and for their family members of any nationality who wish to join them in the UK (see below). Depending on the length of your stay in the UK, EUSS will grant you one of two statuses.
The deadline was 30 June 2021 but the Home Office is clear that some late applications will be accepted. According to its Guidance to caseworkers, reasonable grounds for failing to meet the deadline may include illness, vulnerability, digital illiteracy, or a history of abuse (p26-p40). In fact, in an update from 01 July 2021 the Home Office noted that it "will take a flexible and pragmatic approach to accepting late applications and will look for reasons to grant applications, not to refuse them". We just don't know when this flexibility will end. So please apply as soon as possible.
The main way to apply is online - though you can obtain a paper form in special circumstances (such as being a Zambrano carer), by calling the EU Settlement Resolution Centre at 0300 123 7379. The application will ask for information about your identity, residence in the UK, criminal record, and - if you are not an EU/EEA/Swiss citizen - your relationship to one.
If you've missed the deadline, include a short cover letter outlining the reasons for your delay. For the identity-checking section of the application, you can use the EU Exit: ID document check app, which is available on Android and iPhone and allows you to scan either a biometric passport or national identity card.
Check your email regularly - including your spam and junk folders. The Home Office may contact you to request further evidence, to provide you with a Certificate of Application, or to notify you of the outcome of your application.
Apply anyway. You will need to include a cover letter to explain why you couldn't renew your ID, and demonstrate that you have attempted to do so. This could be a letter, email or a screenshot from your consulate, showing that you were unable to book an appointment to renew your ID in time.
Paper applications could once be downloaded online and submitted via email. As of 23:59 on 05 July 2021, this is no longer the case.
You can apply to switch to settled status as soon as you are eligible (usually after you have lived in the UK, the Channel Islands or the Isle of Man for five years in a row).To do this, you must apply to the EUSS again before your pre-settled status expires. You can find more information on that process here.
Please remember: it is the length of your continuous residence, not the time elapsed since you were granted pre-settled status, which makes you eligible for settled status.