When did the Church of England start letting priests marry?

When Henry VIII started the CofE they were essentially Catholics that didn’t recognize the Pope, right? So when did they abandon clerical celibacy? When did they switch from Latin to English?

He didn’t. He took the existing national church structure and declared it independent of Papal intervention.

Check.

Some priests and bishops, including Cranmer, married secretly under Henry. Most of the freeing from clerical celibacy came about under Edward VI shortly after Henry’s death.

  1. First Book of Common Prayer.

King Henry never accepted marriage of the clergy as summarised by this web-site on the Thirty Nine Articles, a summary of the Anglican doctrine:

Things changed under Elizabeth. In 1571, Parliament passed the Thirty-Nine Articles in essentially their final form. Article XXXII dealt with the marriage of the clergy:

So, from 1571 onwards, there’s not been any requirement for celibacy in the Anglican church.
Oh, and a minor point from the OP:

The Anglican position is that Henry didn’t start the Church of England. He broke off from Rome, but the Church of England was started by S. Augustine’s mission. There has been continuity throughout that time.

Edward and his advisors published a new set of articles in 1553, the Forty-Two Articles, but it doesn’t look as if they were ever official:

Once Mary came to the throne, the Church went back to Catholic doctrine:

So, I would submit that the definitive change was under Elizabeth.