So were there many Roman citizens in Britannia, or was it a pretty small ratio of Romans to locals?

Relevant detail: this changed a lot in 212.

Before that date, Roman citizenship basically implied Roman culture, language and lifestyle; but in that year Caracalla passed an edict granting citizenship to all free men in the Empire, so a lot of non-Latin-speaking locals were to be considered Roman citizens. (And taxed as such).

That said, I'd estimate the ratio of Latin speakers in the province to be 3~6% in the 4th century, based on a few Wikipedia numbers:

• Roman army, family, dependents: 125k people. Likely 100% Latin speakers. You also get a few bureaucrats but they're numerically insignificant.
• Urban population: 240k people, including the above. The others were likely a mix of Brittonic and Latin speakers.
• Total population: 3.6 million people. Unless urban, likely to be Brittonic speakers.

Did the Roman soldiers give commands to the local elites, who would then tell the locals what to do?

Not quite. The army was responsible for the enforcement of the rules, but the ones commanding the local elites and the army were former consuls appointed as governors.

And would you say that life changed much for the locals under the new rule?

I'm not sure at all. But I guess that, for both the slaves and the general working class, there was barely a difference. You still work to the bone, and die an ungrateful death, no matter if you're doing it for the sake of a local tribal chief or for some "imperator" in the middle of nowhere.