If it is the claimant's first WCA, the claimant can self-certify for the first 7 days but will then need to provide continuous 'fit notes' until a decision is made on whether they have a Limited Capability for Work.
In practice, claimants are generally not referred for a WCA until their medical evidence has covered 28 days.
When a claimant first notifies their work coach that they are unwell (ie. self-certifies / provides fit notes), the work coach will generally suspend any work search requirements for 14 days, extendable to 28 days - on the assumption that the illness is just temporary.
DWP guidance in the Parliamentary Deposited Paper on
Work Capability Assessments indicates that in most cases the referral for a Work Capability Assessment should not be made until after 28 days. The only exceptions (in the guidance) are where the claimant fits the circumstances where they can be treated as having a limited capability for work and work related activities - eg. undergoing chemotherapy or radiotherapy, or they fall under the special rules for terminal illness. But there is nothing in the UC Regulations (Universal Credit Regulation 41) to say the referral cannot be made straight away!
If the claimant already has LCW and is requesting a review WCA because their health has deteriorated (see this page for advice on the risks) they do not need to provide fit notes (the DWP has already accepted that they have a Limited Capability for Work). A declaration of how their existing condition has worsened should be sufficient to start a review WCA. However, if they have been diagnosed with a new condition, it would be helpful to provide written medical evidence of that (eg a fit note or a letter from a medical professional).