Biology, Chemistry, and Biochemistry are all pretty distinct degree courses. Most Biochemistry courses have more in common with a Biology degree (and A-level Biology) in terms of "fundamental biosciences" i.e. molecular and cell biology, genetics, biochemistry as it relates to protein structure and function, physiology (especially metabolism) and so on. On the other hand, most degrees denoted "Biology" tend to skew towards the ecological/organismal side than the biomolecular sciences side, although inevitably you will cover some of the same core material.
If you know your interest in the biosciences is on the "fundamental" side as described above, but you aren't sure if you want a more chemically oriented course or not, you may want to consider a joint/combined course in Chemistry and/with Biochemistry (or vice versa), or a Biochemistry course which has a large core chemistry content (like at Oxford), if not a Natural Sciences course. "Biology" as above, where not otherwise specified further, tends to focus more on the ecological/organismal side of things - you may want to compare the course in Biology and Biochemistry at e.g. Southampton and Oxford to see the different perspectives the courses typically have.
Chemical Engineering has very little in common with even a Chemistry degree, much less a Biology degree. The above advice regarding engineering generally applies equally to ChemE. There are a small number of ChemE courses which allow you to pursue chemistry as a "minor" subject (usually denoted "Chemical Engineering with [Medicinal/Applied/Industrial] Chemistry") however, but in general ChemE have more in common with Engineering broadly, or perhaps (Applied) Mathematics, than with Chemistry proper.