There is a set of research guides produced by the UK's National Archives (TNA). The guides you need are accessible from the "Looking for a Person" page - right-hand column, look for "Criminals, bankrupts and litigants". Essentially, the location of any surviving records will depend on what level John Edmonds was tried at. TNA at Kew holds the higher levels of courts and potentially calendars of prisoners. The records of such a trial are, I suspect, more likely to be found in Petty or Quarter Sessions courts, which will be held in country record offices - so you'd need to know which county. Some further references to Welsh courts can be found on this page. (The Welsh legal system was part of the English, unlike the Scots' system which was preserved as a separate system. Nonetheless records of Welsh courts will be found in Welsh record offices).
The warning on the TNA page "Criminal transportees: further research" says
Few records survive about individual convicts who were transported to
North America and the West Indies.
From 1718 to 1776 transportation was entirely to North America.
After 1776 no convicts were sent to North America. The legal records
discussed in section 5 that cover dates before 1776 may contain
material on transportees shipped across the Atlantic but they are
significantly less numerous than records of convicts sent to
So you may be lucky but ...
There is a book "The complete book of emigrants in bondage 1614-1775", followed by "More Emigrants in Bondage, 1614-1775", which contain about as many names as are known. The latter book is on Ancestry but not the earlier, so far as I can see.
In fact, the books were superseded in 2005 by a CD "British Emigrants in Bondage, 1614-1788", author (as before) Peter Wilson Coldham, published by Baltimore Genealogical Publishing Co.