Yes, this is Steatoda nobilis, the so-called 'noble false widow', the spider which the media make such an unjustified fuss about. It has been frequent in the southern coastal counties of England for a great many years, without causing any problems. It is nowadays frequent and abundant in many areas of southern England. You can see current distribution at the summary page for the spider. The media frenzy caused by this spider is unwarranted. There are a number of other 'false widows' in Britain, some which are native or have been in the country for centuries, so there is a lot of misinformation about these spiders. Steatoda nobilis spiders are unobtrusive, have no interest in attacking humans and would only bite if they were put into a position where they do so as a defensive reaction. Honey bees and social wasps pose a far greater risk.
Thanks for the details.