Hi Llamafarmer
Searching for living persons is always a bit hit-and-miss even with an unusual surname. But to search for a SMITH without a middle initial would be daunting to say the least.
If we look at the years 1932 - 1936, the most probable time that your William Smith married, there are 66 William Smiths (14 with no middle initial) listed as registered in Birmingham. Not having used the GRO or BRO search service, I don't know what depth of search they are willing to undertake - maybe indexes only which would be of little use to you without a spouse name. Best to give John Yates a ring at BRO and explain your requirements.
But don't despair - I've had some great successes in tracking down children, grandchildren, brothers and sisters for people on this and other sites. It is time-consuming but very rewarding if you are prepared to stick at it.
If any of your Smiths have middle names, then your chances of success are much greater.