I'm less familiar with the details of the British evacuees. But I can discuss a little about the French.

Roughly 140,000 French troops were evacuated along with the British. These were set up in a series of quickly erected tent camps in the southwest of England The largest of these was Tidworth in Wiltshire, where the British barracks were lent to the French and a large tent camp was attached.

The French had three priorities, to supply these men, to reorganize them, and to get them back in the field. Within a week they started repatriation. So while some ships were still running the gauntlet at Dunkirk, others had started shuffling troops back to France. There were even a handful that shuttled men directly from Dunkirk to Cherbourg, never seeing England at all.

The evacuated units had lost all their heavier equipment (trucks, artilery, etc) and most men had lost their personal equipment. So while the French had the men, they had to be re-equipped, generally in an ad hoc way. Most of the 140,000 (112,000 or so) French had made it back to France, but only about half of them were properly reformed into functional units. Some of these were basically "new" units, the old 1st and 2nd DCR divisions were disbanded and formed anew. While other existing units were reinforced. de Gaulle's 4th DCR, for example, was brought back up to full strength with Dunkirk evacuees. There men from Dunkirk at all the major battles after Dunkirk.

After the fall of France roughly 25,000 French soldiers still hadn't been repatriated. Of these most were repatriated to occupied France after the ceasefire. About 3,000 joined (and basically started) the Free French under de Gaulle. And perhaps a couple thousand simply deserted, stayed in England and become for all intents English. Although ultimately all the numbers are guesses. Good records were not kept in the tumult of that couple of weeks.