My advice...
Depending on the day of the week, and time of the year (although maybe not as much of an issue mid-January as in the summer) you may find that the Register Office has multiple marriages and/or civil partnerships on the same day.
Unlike a wedding service at a church this could result in a wedding party arriving every 15-30 minutes. The register office in Portsmouth, for example, can do 7-10 weddings on a busy day. That results in wedding parties arriving, entering, having photos taken, and leaving in a short space of time and you could find 3-4 couples "onsite" at any one time. Not only might this scupper you planned place to take any formal photographs but it will almost certainly involve you ensuring you carefully select the subjects in your images, and your composition.
Nothing worse than delivering a set of proofs containing other B&Gs guests be they formals or candid images. I know photographers who have done this (although not me when I was shooting them, and fortunately I only had a few of them in my first couple of seasons).
So it might be worth, a week before, calling up and seeing what the timetable is - you might be lucky and be the only one on that day.