Nothing will happen. The various governments will agree on keeping the existing arrangements as otherwise it will be too complicated and too many companies will lose out, with their staff suffering too; there is too much tax revenue involved between corporation and staff income taxes to chance individual new negotiations with all the other issues to be dealt with.
Likewise, I'm sure there will be agreement to ensure existing staff at all EU and UK operators are given rights to stay and work in the country they are presently base in as long as they are employed by that company, though those rights would possibly be lost if they leave the company.