Since Diablo 3 doesn’t require a monthly subscription and overall is not so complex as World of Warcraft, there is no Customer Support in-game like we have in WoW.
There is a Customer Support button in the menu but it is leading to a webpage, http://eu.battle.net/support/en/games/diablo3.
This page has a list with articles regarding common issues with Diablo 3, you can even submit a ticket from there after selecting various titles and categories which may lead to a dead end, it will say that you can’t contact them for some specific issues.
You can call Blizzard by phone, though. All the available phone numbers are listed on this page. They will answer from 9:30am until 10pm Paris Time, 7 days a week (Europe).
For UK: 0800 0288246
Some countries are not in the list. You can still call Blizzard by using Skype, you can dial the UK number in Skype. They are offering support for all the Blizzard games, WoW, Diablo or Starcraft.
The numbers starting with “0800″ are free numbers, Skype won’t charge or request any money for the calls.