I have never felt that my baggage, when tagged as priority by BA, regardless of travel class, ever arrived as a priority. There are other airlines that I know for sure that my bags will be out first, but with BA it is always pot luck as HongKongLady also mentions.
In terms of whether the bags for Silver/Gold should be tagged – I find that to be at the discretion of the staff at check in. There does not seem to be anything mentioned in the benefits section for Gold/Silver other than the following for Silver and similar for Gold:
“Leave the queues behind
Your Silver membership opens up access to worldwide priority check-in desks, priority boarding and other services, whatever cabin you’re flying in. As a result, you can spend more time in the lounge or on the plane and less time standing in line.
•Access to Club World, Club Europe and equivalent oneworld business-class check-in desks
•Priority at Baggage Arrival Services desks
•Priority over Blue and Bronze Members on reservation waitlists and airport standby lists”