I think the visa process breaks you of sentimentality over the traditional things. It becomes an exercise in practicality to some extent as you have to forego the typical relationship patterns and timings that others may get to choose including living together, where and how to marry and when to take a honeymoon.
By all means, make your legal ceremony mean something special and sentimental to the two of you. Or make plans to pretend it never happened and don't celebrate it. But I would encourage you not to jeopardise your visas and stay in the UK for the sake of moving the legalities forward by 2-5 weeks. If the worst case were to happen and your partner couldn't get a job in time you face spending a not insignificant percentage of your first year married living thousands of miles apart.
Larissa has hit the nail on the head.
I think a lot of us have (unfortunately) been "toughened up" because of immigration laws. I know my husband and I simply eloped as:
1. We couldn't be bothered with the amount of hoops that had to be jumped through to marry in the UK
2. We wanted our ceremony to be how we wanted it (my brother officiating, outside, all our family) and that wasn't possible
3. No matter what we did, one of us wouldn't have a lot of loved ones there
So we said "screw it" and just went to NYC and married just the two of us. We have no regrets and we love our story. Especially when you heard wedding nightmare stories.

But I completely agree with Larissa. Decide how best to approach moving the legal bit up ASAP. There is likely a compromise in there somewhere. You can do this! I'm glad it's not cold feet and is sentimentality. Sorry for jumping to the cold feet bit. I forget that sometimes men want the white wedding as much as some women do.