Applying for a postdoctoral fellowship in the UK, how can I find a host institution
In general, you should reach out to the person you'd like to collaborate with and go from there. Email professors you're interested in - they can help deal with the bureaucracy of their institution (approval to host etc). Bear in mind that some universities set stringent internal deadlines on applying to certain fellowship schemes.
Whether you'll be PI depends on the individual grant scheme. You'll have a line manager (most likely the professor you apply with) but not necessarily a supervisor-supervisee relationship.