To some extent this will be for the UK government to decide. However, the UK government wants maximised access to trade, and the other states (particularly the Eastern European states) might drive a bargain in that regard and insist on maximised access of their citizens to the United Kingdom.
Thus, it depends partly on the negotiations and partly on the seriousness of the UK government’s interest in imposing restrictions. Restrictions are something of a political imperative, but the economic imperative is pointing in the other direction. Obviously, if restrictions such as visa requirements do happen, it will increase the bureaucracy and the time involved for all concerned. It does underline the point that Brexit negotiations are strange negotiations: usually, when countries are negotiating in relation to trade, they try to increase the trade between themselves, but we are currently in the position of maximised trade and maximised free movement of persons between the UK and the other 27 member states. Ironically, where we are going to end up as a result of the Brexit negotiations is less trade between the UK and the rest of Europe and, perhaps, less migration too.