Why would they elope to Wick?

It's as far as you can go while remaining in the UK

Wick is at the end of the Far North Line in Scotland, UK, making it the furthest railway destination that Gary could select.

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Miranda and Gary have both tried to break through their fears and inhibitions to find love with one another, and repeatedly been thwarted. Gary reasons that by travelling as far away as possible, away from pressure from family and friends, an elopement will enable them to finally cement their relationship. Miranda, however, has worked through her issues by this point and is ready to marry Gary at the restaurant in the midst of her social set and relations.

Miranda in turn bestows the same gift of freedom on Tilly, presenting her with the tickets and the instruction to elope with 'Dreamboat' Charlie.

MIRANDA: I don't need to elope, it can be on my own terms.
MIRANDA (to Tilly): You have these. Be free of your mother.
'DREAMBOAT' CHARLIE: I bloody love Wick!

As for the real-world legality of these arrangements...

Historically Scotland has had more liberal marriage laws than England. In particular following the 1754 Marriage Act, young couples (under 21) would head across the border to places such as Gretna Green where they could marry without their parents' consent. Scottish law at one time allowed for "irregular marriages", meaning that if a declaration was made before two witnesses, almost anybody had the authority to conduct the marriage ceremony.

However subsequent reforms brought Scotland's law closer to England's and today the requirement for 28 days of notice to marry is universal in the UK.

In any event Miranda's request to the registrar to marry her and Gary back at the restaurant immediately would be illegal and irregular as no notice had been given. Even if they had gone to Wick they would have had to wait for the 28 days of notice to elapse. The requirement for notice can only be waived if illness of one of the partners or somebody important to the wedding means that the 28 day period poses a problem. To get married quickly Miranda and Gary would have to head to another jurisdiction such as Las Vegas.