This should be discussed with both the Exec team and with People Ops before any action is taken in order to ensure it is done in compliance with local laws and regulation. As soon as you know you'll have to let someone go, do it immediately and with respect.
The team member is entitled to know where they stand. Delaying it for days or weeks causes problems with confidentiality (finding out that they will be let go), causation (attributing it to another reason), and escalation (the working relationship is probably going downhill).
The People Ops person, the manager and their manager gather the PIP results, and share that with the team member in a closed door meeting at the end of the day. The news is shared. The team member is asked if they have any disagreements on the decision. If there are none, they are told that they have the option to resign, as opposed to being terminated. The rest of the organisation will be told that they are resigning, so that this data stays with us. This is done to ensure the exit is respectful, the person feels they can continue to honour the friendships they have made in their stay in the organisation without any undue negative feelings, and their future employment options are not hampered.
A date for the exit is agreed upon (if it's an exit owing to underperformance, the organisation can decide on an early exit date, serving of the notice period can be waived off), the person sends in their resignation on email within the next 3 working days, addressed to the people in this meeting.
All other practices related to exits, such as a farewell breakfast with the team, a keepsake, a speedy full-and-final settlement, etc. will all be followed.