For entry into the Schengen Area (not within the Common Travel Area), British citizens will be treated as 3 Country nationals.
A entry/exit stamp will be placed in their passport to enforce the 90/180 days rule.
An exception will be made for those presently (2020) inside the Schengen Area. The 90/180 day clock will start on the 2021-01-01, so the lack of an entry stamp will not automatically assume an overstay.
Should, for any reason, no stamp upon entry be given it is advised to retain some proof of date of entry (ticket etc.) to avoid complications when leaving.
The same is true for (non-Irish) EU Citizens when entering the United Kingdom, especially when arriving from the Republic of Ireland. At present it is not clear how the date of entry will be noted for those travelling with only an ID (which will still be possible until 2021-10-01), afterwhich a passport is required for non-resident EU Citizens who are not Irish.
As to how private corporations, who will be held responsible for transporting persons without the proper paperwork, will deal with this is their affair. Assume they will refuse to allow you to board if you don’t fulfill their conditions.