I really enjoyed my research project and want to do a PhD.
I would like to take on an oncology or neuroscience PhD but I don't know what areas within these general themes. How much of the field will I be expected to know before I apply? How can I find clear and concise information on research in these areas?

Searching this site will give you plenty of specific projects relating to biochemical aspects of oncology, neuroscience or neuro-oncology. Research at this level however doesn't easily fit into concise summaries. You just have to wade through all the things on offer until you find one of interest. Its something you'll need to get used to doing once you start your PhD.

If you pick a project which has a very biochemical leaning, your supervisor may consider your relative weakness in neuroscience or oncology as a fair price to pay for a good biochemist. Your first degree should have covered at least some neuroscience or oncology. If you’re offered an interview then read any reviews you can find. No supervisor expects their student to start off as an expert; a capacity to learn and understand quickly is what they are looking for.