If I have a standard home Broadband connection, in the UK, which I can connect any device to wirelessly to and it appears to connect OK, and no websites appear to be blocked, but my XBOX 360 connects to the WLAN but does not connect to XBOX Live, what could cause this?
Don't forget... There is a difference between connecting to Wi-Fi and having Internet... Your WiFi network could possibly lose its DNS, or lose Internet Connection... Connecting to the Wi-Fi is just one step of getting onto internet.
Make sure with another device that you can both get onto the Wi-Fi, and that it can browse the internet. If that works, on your Xbox, try to access the internet (not get on live, but browse the Internet).
After that, you know the problem is somewhere between your Xbox 360 and Xbox Live or that you have ports blocked, via a firewall or by ISP provider, or if you did a factory reset of your router recently it maybe the problem. If you haven't touched your router lately, and you are 100% sure, it might be the Xbox Live Server in which you text/call/ask friends or google it and I'm sure millions have complained about it already :)
I had the same problem this past Saturday, I was unable to connect to Xbox Live, but when testing the connection it said it could connect to Xbox Live and there were no Xbox Live services issues reported.
I went through the Xbox support page's list of recommendations and nothing helped.
What ended up working for me was rebooting my router.