I was going to suggest Scotland, but I thought that was because the age of consent was different. In the UK you can get married at 16 with parents' consent but not until 18 without parents' consent, but in Scotland it's just a flat 16, so couples where one was 16 or 17 who wanted to marry without parental consent went to Scotland. There are other scenarios where people went to Scotland to get married but I can't remember what they are.
The above info may well be out of date by now.
Trying to bribe public officials is a) difficult and b) illegal, with significant jail terms. I mean you could in theory bribe a public official to do anything but it would be considered a criminal act, not a favour for a friend.
Does it have to be a legal wedding? You can have a religious only ceremony without giving any notice at all. You won't get the legal paperwork but you'd still get a wedding. Mainstream churches won't do that but plenty of other people do. It used to be the case that only churches and registry offices (possibly synagogues?) could do legal weddings. People from other religions would do their own weddings on their own terms and then people would later sort out the legal paperwork with the most basic registry office wedding, but would consider themselves married when they did the religious ceremony. I know that some masjids/mosques would do weddings at short notice, but it's only recently that masjids/mosques have been allowed to do the legal side of the weddings. (My ex partner is a Muslim so this isn't just speculation.) The limitation with this is that religious establishments generally only provide these services for members. Like if you wanted a quick Islamic wedding, you'd have to convert to Islam first, but you will find imams willing to do this, as long as you're sincere in your desire to become Muslim and have a Muslim wedding. That probably won't help your characters if they're not Muslims or don't want to become Muslim, but it's an example of how non-legal religious only weddings aren't exactly rare. Your characters may find some small, liberal-minded church that will do a non-legal wedding, although they may call it "blessing ceremony"* - I know the Unitarians used to do this for gay couples before gay marriage was legal. The minister will make it very clear that it's not going to come with the legal paperwork.
*they don't have to call it that, they can call it a wedding if they want.
Some of the above may have changed since the law changed to allow a wider variety of places and people to conduct legal weddings.