We will have friends over visiting us on Schengen visa. We are considering going on a weekend trip to Kent. They would need a UK visa to enter. How quickly can this be arranged and is it possible close to the border, or do we need to plan a visit to the consulate in Brussels?
Asking for visa in their home country is not an option anymore.
Bad news - the answer appears to be no.
The British Embassy in Paris website lists their locations as Paris, with consulates in Bordeaux and Marseille and a British Trade and Investment Office in Lyon. It makes no mention of anything in Nord Pas de Calais, so it would seem that the previous consulate in Lille (for which there are references online) has closed. The British Embassy Brussels site makes no mention of other centres. Luxembourg appears to be handled through France, so no option there either.
Both of these link back to the new-ish UK Visa Application Centres page, which takes you to here for France and here for Belgium. It seems this has all fairly recently been outsourced, and as is often the case for this sort of thing, made more useless and less helpful. The only two locations currently listed as being available for visa interviews and issuing are given as Paris and Brussels.
So, assuming the Do You Need a Visa wizard comes back as saying your friends really do need a UK visa, the only option appears to be applying online, and heading to either Brussels or Paris for an interview, assuming one is available in time.
