Depends on how many guests you want there, I think.
Perfectly possible to have a small (lovely) wedding on the cheap.
We had a tiny wedding in a registry office. I got a a dress in a sale, DH had a copy of his favourite designer suit made by a tailor.
Taxis to a restaurant afterwards.
Because of the small number of guests, we could afford to book lunch in a private area of really beautiful, top notch restaurant in a lovely part of London.
Afterwards, we had more champagne in a lovely bar with views nearby, then DH and I got taxis straight off to our honeymoon.
The wedding itself cost £1,500. Honeymoon £3k.
We had a big party in a bar when we came back, which cost under a grand.
Its the sit down meal feeding the five thousand and all the faff - cars, flowers, bridesmaids etc - that cost.
You can cut all that out easily.