Get a price so as to compare with a local tailor.
Having Oxxford doing the lining is similar to "recrafting" by the shoe manufacturer. You might pay more, but if you will get the same workmanship, details, etc. You might get an overhaul.
If you have a quarter lined coat, I would go to Oxxford. An outside tailor may use shortcuts. E.g., machine finishing the lining rather than hand sewing. Also, an outside tailor might not get the bellows pockets right.