This is, it seems, a Welsh expression. As a native of Wales, living in New York, with a son visiting from Texas, my husband happened to use the phrase "he'll be in his oils" referring to a fellow Texan. Not knowing where he had heard this first, (and thinking it had something to do with the Clampetts) I googled it. Actually, I used the phrase "in one's oils' (because I'm posh now!) which returned this explanation (one of only 8 results, and the most plausible)
"In his oils - Listed and explained in the Talk Tidy glossary [by John Edwards] but I omitted to show its Welsh origin for 'in his element'. The original Welsh expression is 'Yn ei hwyliau' which, via mutations like 'in his hwyls' became finally 'in his oils'."