As much as we'd like to, we can't hold your hand and run through every element of the job unfortunately, so done of it you have to find out for yourself. A bit hard I know, but it's your responsibility to comply, not ours.
Local Authority Building Control departments are being squeezed but on the whole trying their best to provide the best service they can, mostly with declining staff numbers.
I was on a seminar last week & it's surprising how many authorities are stuck in the past & very slow to embrace change & new developments. Very surprising that this was the first stove app they'd dealt with, suspect there will be hundreds of non compliant installations on their patch which they should have controlled.
Flue test/check new, possible air vent, hearth/surround check, C0 detector, job done.