In theory, items left on a train will be carried to the final destination station and the operator of this station, regardless of the operator of the train, will be responsible for handling any property that is recovered.
XC▸ are rather good at passing the buck when it comes to lost property as they don't operate any stations and as such they don't handle any such items themselves. If you leave something on a south bound XC service their stock response will generally just be "speak to
GWR▸ " as the train will have terminated at a GWR station. What they don't take into consideration however is that XC are responsible for cleaning their own trains so GWR will only receive items if XC find them and hand them over. If XC don't clean their train, or if they don't find the item in question (maybe it's in the overhead rack and the cleaner isn't tall enough to see it), or if they don't/can't hand items over, then there is no way that GWR will have them.
This particular train arrives into Paignton at 1855 and moves off to the sidings shortly after. At around 1940 it pulls back into the station in time to form the 2014 departure to Birmingham New Street. The train will be cleaned by the on board crew while in the sidings. During the day any items they find will be handed over to station staff, however, the station staff go home at 1900 so there will be nobody to hand any items to at this time. Therefore it is most likely that if these headphones were found on the train they would have remained onboard and returned to Birmingham with the train.
Try contacting the Excess Baggage Company as they handle the lost property at Birmingham New Street.
www.lostproperty.org or 03300 240 215.
Hope this helps.