Overall process from my experience is your supervisor speaks with you and informs you your being surplused and you have 60 days to find a new job in the company. You then get a formal 60 day letter from HR.
The specific day can vary and each group and supervisor can choose to handle the specifics differently, but they have a scripted procedure your supervisor is told to follow.
If it's a large group having cuts, then someone from your management chain will setup group calls from those impacted and those not. In some cases your asked to join a call and told your whole group is being cut.
Used to be the 1st month in a qtr is when you had to worry. That would allow 60 days, but have the people off payroll by the end of the qtr.
Chances of finding a new job can vary a lot depending on location, your experience, number of others we also got 60 day letters or if there are really any jobs even available. In one case they shut down the system for searching for new jobs during those 60 days.
During your last 60 days people are expected to continue doing their old job, but it's understood your job search is your main focus.