Church of England (which included Wales then) baptism records won't normally give the mother's maiden name, so not helpful.
Non-conformist baptisms, you'd have to find it first.
You need a birth cert for one of the kids to get the mother. Emily's would be the easiest to pick in the index. From the census image, she was 10 months old in April 1851, not 1 month as given on FamilySearch.
Given the mother's name, you can probably pick the marriage. The marriage cert would hopefully give both fathers' names, though sometimes they disappoint. You might not need it for Anne's father if you can find her in 1841, though 1841 doesn't help with William.
There are normally 3 marriage records
(1) signed on the day, held by the main register office that now covers the location, though some districts have moved them to the Archives. Charges vary. What you get varies - some places can send an image, other places just copy it out. Indexing varies - some places can't find the record if you don't know the church.
(2) C of E weddings only - duplicate register, also signed on the day, held by the diocese, somewhere. Often a county archive, though maybe in a different county. These registers are often more accessible. They've often been microfilmed along with the baptisms and burials, and may have been transcribed by the LDS or the local FHS. Or somebody local might do you a lookup. It's not unheard of for the Archive staff to do a lookup or supply a cheap photocopy, though others will say they aren't allowed to. But the big problem again is that there might be 30 churches in a district and no index.
(3) the GRO copy. These are transcripts made out at the end of the quarter by the vicar or registrar. The advantage is the indexing. But sadly their index refs are no use for the other registers.
