The marriage certificate will give you their fathers' names and occupations. I would think you are certain to get the right birth certificate for Walter anyway because of his middle name, but if you had Elizabeth's father's name before ordering her birth certificate, then it might be easier to make sure you order the right birth certificate because we should be able to find her family on the censuses, and the family of the "other" Elizabeth, and hopefully work out from the census info which is the right birth registration.
Fingers crossed the birth cert you've ordered turns out to be the right one anyway!
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