The obvious answer would be to refer to them as King Edward, and Edward II respectively. Their epithets were created after their deaths and they were canonized as saints.
But King Edward I is Edward Longshanks. And Edward II is his son. I realize that the former Edwards were Anglo-Saxon kings, while the latter pair were Normans, but the crown, sword, and staff of St. Edward(the Confessor) are used in Coronation ceremonies.
Technically, those are replicas, since the originals were destroyed during the English Civil War. But those replicas were created circa 1660, and named for Edward the Confessor. Until they were destroyed, even during the reign of Edward Longshanks, the original St Edward's Crown was kept as a holy relic at Westminster.
So why is the third King named Edward the First? And what did they call Edward the Confessor while he was alive?
In short answer, the ruler name with regnal number like "Edward I" was much less common in contemporary (medieval) documents at least by the end of the 12th century than generally assumed.
As for Edward I "Longshanks", this post by /u/Valkine offers a useful and detailed insight on the topic in general as well as the specific circumstances in the 13th and 14th centuries.
And what did they call Edward the Confessor while he was alive?
This is much easier to answer (even for me).
The followings are the links of the documents issued by individual rulers under E, including both Edwards, compiled in Electoric Sawyer, the most famous database collection of pre-Conquest documentary sources in England so that you can also check by yourself: https://esawyer.lib.cam.ac.uk/browse/kinga/E.html