It helps when you don't have computer records. Barry was of little signficance when he deserted.
It's not "Sci-Fi", it's "SF"!
"Calvinism is a very liberal religious ethos." - Truekiwijoker

He clearly deserted from his regiment and when he enlisted
Why was he never charged?
He was caught, by a representative of Prussia, Britain's then ally, by Captain Potzdorf, who offered him the 'choice' between re-joining the military, in this case the Prussian army, or being 'given up', that is, handed over to the British and charged with desertion. Barry 'chose' the former. All of this occurred on the Continent, on mainland Europe, during the Seven Years War of the 1750s, in some unspecified (in the film) place, whether France, Germany/Prussia, Austria or elsewhere. When he eventually escaped to England some time later via the Chevalier, nobody there would have known about his past and nobody would be looking for him, either there or elsewhere. Besides, those who did know about his original desertion would likely by then have been all dead (via battle, disease, etc), as back then mortality rates in war and battle were close to 100 percent.
Additionally, many years had passed. Barry marries Lady Lyndon, as the narrator of the film indicates, in 1773, long after the Seven Years War, which had ended in the early 1760s. Lord Bullingdon (an aristocratic title, not a name; his surname is Lyndon) would have known nothing about Barry's military past.