Integrated masters to PhD?

Yes, either would be fine. An "honours degree" is any classed (i.e. 2:1, 1st, etc) undergraduate degree (i.e. not an ordinary degree, which you can't usually apply to directly and is just offered as an exit award if you fail certain components of the degree, like the dissertation...in which case a PhD isn't happening anyway usually). The fact it's an undergraduate masters course does not change that. It's very typical for people to go from MSci/etc courses to PhDs directly, in STEM fields at least.

Note that "biochemistry" is not the same as "biology and chemistry". A degree in biochemistry is fundamentally a bioscience degree focusing on molecular biology. There will be some ancillary chemistry content but 90% of the course is going to be biology. If you want to continue studying chemistry in a meaningful way past A-level you should look at chemistry degrees (including joint honours courses).

Also as an aside, Warwick isn't especially well known for biosciences or chemistry. Their flagship department is maths, and their other standout departments are the business school, economics, and CS, plus somewhat notable offerings in classical archaeology and art history.

Is Durham good for Biology and/or chemistry?

Is Durham good for Biology and/or chemistry?