On the home page, the “How it works” three-page widget scrolls too slowly when I click one of the three page circles. That speed is fine for the automatic page-turning every 8 seconds, but if I click a circle, I want to see that page right away – the animation should be about 5 times faster. I was slightly annoyed by the slowness when I tried to go back to the first page, which you don’t want.
It also might help to stop that widget from scrolling until the user has scrolled far down enough that you think they have started reading it, so they can read the first step first, and aren’t forced to click on the first-page circle to go there themselves. Perhaps you don’t care which page the user reads first, but the chronological ordering of the pages contradicts that idea.
When I read https://www.sleepio.com/sciencebehindsleepio, I was impressed that you said you had done experimental trials, and I couldn’t see any sign of obvious bias in the experimental design, but I was still a bit skeptical and worried that the experiments were biased. I just realized that an addition that would have convinced me a lot more strongly would have been a LaTeX-formatted scientific paper about the experiment, typeset in Computer Modern. The paper would not promote Sleepio, but just describe the experimental method and results. I don’t know if LaTeX supports exporting to a web page with the Computer Modern font embedded – if not, you’d have to link to a PDF, which you’d have to work hard to convince me to open.