British citizens don't require a visa to enter into Norway, and thus Svalbard because Britain is a EU member country.
Citizens of Countries with Visa Exemption
All EU or Schengen countries or countries with whom Norway has entered a visa exemption agreement, can enter into Norway and thus Svalbard without obtaining any visa. Visit the Norwegian Directorate of Immigration Website to check all these countries.
Citizens of Countries who require a visa
Nationals of all remaining countries require a Schengen Visa to enter into Norway and thus Svalbard.
You have to register yourself on the website (portal) of Norwegian directorate of immigration at the link to get a Schengen visa to enter Norway and thus Svalbard.
Some generic documents that may be required are mentioned below:
a) A valid passport
b) Proof of Funds to finance your stay
c) Visa Fee
d) Completely filled and signed application form
Submission of Visa Application
You have to personally visit the Embassy of Norway in London, and register your bio-metric data (10 fingerprints and digital facial image), to apply for a visa to Norway (thus Svalbard).
Visa Processing time and fees
It is suggested that you apply several weeks before your visit but usually it takes around 10 days to issue a visa.
The fee for a Schengen visa can be range between 30EUR to 70 EUR depending upon classifications or categories. Please visit the Embassy of Norway in UK Link to check the updated fees and processing time.
Collection of visa
After you have applied; you will be required to pick your passport from the Embassy of Norway in UK, at a given time duration.