How to fix a messed up replication on MS SQL Server
TLDR:

It appears that disabling and re-enabling replication probably fixed the issue:

exec sp_replicationdboption @dbname = N'DatabaseName', @optname = N'publish', @value = N'false'
exec sp_replicationdboption @dbname = N'DatabaseName', @optname = N'publish', @value = N'true'


I guess this is the equivalent of switching it off and then back on again...

Longer version:

A workmate had a go at trying to fix it. He tried a few things but didn't get very far. The one change he did make before giving up was to disable the replication.

I then tried Cody's suggestion. The sp_dropsubscription command complained that no subscriptions exists. So I tried the sp_droppublication command. This complained that replication was not enabled on the database. So I enabled it and re-ran the command. This time it complained that the publication did not exist. I refreshed the Local Publications node in SSMS and sure enough it had gone. I ran the replication set up script, generated a new snapshot and every thing is now working properly. Joy!

I'm not 100% certain that disabling and enabling the replication is what actually fixed the problem, but it is definitely worth trying if the replication gets messed up.