I found this spider (~10 mm body length) indoors on a wall (South-East England) and I was wondering what species it was.
My first guess is false black widow, but I can't be sure.
I'd be grateful if anyone could help me out.
Yes, it's Steatoda, probably nobilis, the False Widow. The white band curving around the front of the abdomen is a good Steatoda diagnostic, and the light-colored abdominal pattern is pretty variable in this species - only hints of it here, but the pointy front end and the bits of a skeletal 'shield' pattern are there. The spider is pretty harmless, although it is supposed to sometimes give a bite that's like a very mild Widow bite. Not at all dangerous, but a little more potent than the usual harmless spider bite.
Below are a couple of pictures of nobilis, showing a range of markings:
Here is a citation describing the effects of S. nobilis bites that's not a fear-mongering tabloid story:
The images were taken from a random survey of Steatoda nobilis images from the internet, as a broad selection was desired. However, this note (below) discusses some of the features I mention, along with photographs that support my comments.