I remember myself and/or the vicar humming "The Lord is my Shepherd" to my husband to see if he recognised it. He said he did, but on the wedding day he said he must have been mistaken, be didn't know it at all. I have since learned that it is a very English tune. However, after decades of singing in German services, I have discovered that many hymns have the same tune, in fact many of the hymns we sing in CofE services are written by German composers.
I'll never forget watching Princess Anne's (first) wedding with the grandparents of the children I was looking after as an au pair in Mannheim - she had "Glorious things of thee are spoken" sung at the service. (Tune by Haydn). The Germans were amazed, as it is the tune to their national anthem.
Nowadays we sing all sorts of songs and hymns at weddings (I still sing in the church choir) . Popular ones in recent years have been Bette Miller's The Rose, Chris de Burgh "In a country churchyard" and John Legend"All of Me".. We dont expect the congregation to join in as they are not usually church goers and trying to get them to sing is a waste of time.