Not an official answer, but may be of some help:
Currently you can check valid names using the XML gateway and the NameSearch call: http://xmlgw.companieshouse.gov.uk/index.shtml but documentation is a bit all over the place though. This will give you exact matches, but also Company Names that Companies House considers "Same As"
I don't think there is anywhere online to validate against Sensitive Words and Expressions (section 7 in your link). For these you may need to maintain your own database and validate yourself. Although many of these words and expression can be used if you have the appropriate permissions.
For the example you gave, not being able to register Nikes is nothing to do with needing permission, it just fails validation as it is the same as an existing name with S on the end (and UK is ignored for the purpose of naming).
However, there may be some case-by-case manual validation for names that are obvious attempts at passing-off, so "Nike Shoes" might get manually rejected because it's so obvious, but there's no electronic validation for this that I'm aware of, so there's no way to check prior to incorporation. The same is also true for names that may be misleading in other ways. Beyond that, it may become a Trademark/Patent Office issue rather than a Companies House issue.