The Civil Contingencies Act gives the government wide-ranging powers to modify or suspend almost any primary legislation. When the media talk about "Martial Law" what they are most often referring to is the power of the government to declare an emergency, and act without the consent of Parliament for up to 30 days.
The Human Rights Act would remain in force so the protections describe therein would remain. But almost anything else could, in theory at least, be changed.
For an analogy of how things could work out, you could look to the foot-and-mouth outbreak in 2001. The government suspended the right-of-way along most footpaths and "closed" the countryside. Farmers were not allowed to move livestock around, and if foot-and-mouth were discovered, their herd could be appropriated and destroyed. These restrictions were created without primary legislation. Instead, these were temporary actions by the executive, acting under the Royal Prerogative.
In the case of the UK leaving the EU without a deal, emergency action could include the use of the military to control and redirect lorries that would be carrying goods across the channel. Using the military to aid the border force in managing immigration. Using the military for maintaining order.
It is not possible to describe in detail what the exact implication of emergency powers would be on various groups. As it is unknown what exactly would happen, and what the response would be.
It is worth noting that emergency powers can only be introduced for a maximum of 30 days. After that, the changes must either be reversed, or approved by Parliament. And as noted above, the human rights act remains in force, and the judiciary would still function.
Similarly, the effect on the Nationalist movements in Scotland and the North of Ireland is unknown.
