It allows you to access the speed test pages. It does not give access to any other pages. That is its sole prpose, so it should
not allow access to any other web pages.
If you want to access the web, including to run an overnight trace, you need to use an ISP. If you want a speed test independent of your ISP, you need to contact the BT Speed test.
What value do you expect to obtain from an overnight traceroute? It is a very intensive task, placing a far from negligible load on the web as a whole and if your machine is losing its connection, surely you should be looking for something far closer to home and between you and your ISP.
John Underwood