The same thing they do with divorced and re-married Catholics--discern whether the original marriage was valid (annulment). It "does" happen in the RC church.
As to what happens to the married Anglican priest--that depends on what HE wants. Quite a few married Anglican priests have converted, and are now Roman Catholic priests (and still married). I think the only caveat is that if his wife dies, he cannot marry again. And, or course, the church has to agree to accept him. But, in any event, the precedents are already in place.
I thought it only happened in the RC church if an annulment was granted? No way am I going through that; I don't believe in it.
Married Anglican priests have remained married and become Catholic priests. I believe we have two in our Archdiocese.
I would not want to touch the divorce/annulment issues at this time.
I know that the church has done that in the past with Anglican priests, but I thought that the Church did not accept the validity of Anglican Sacrament of Orders. Do you know if an Anglican Priest converting to Catholicism must be re-ordained by a Catholic bishop in order to function as a priest?