No player shall be allowed to play for more than one Sunday Adult club. (similar rule for Saturday adult clubs too)
So, from the original post, if a player has played for Wyvern in County Competitions this season he will be cup tied and therefore ineligible to play for Birstall Rangers
If a club has two (or more) teams, a player can play up in a competition (played for the Reserves, he is eligible for the first team), but cannot play down (having played for the first team, he is not eligible to play for the Reserves)