You need to look at the GPDO 2015 as amended. It falls under Class E (being a building incidental to the enjoyment of a dwelling house).
Para E.1 (e) states that development is not PD where:
e) the height of the building, enclosure or container would exceed—
(i) 4 metres in the case of a building with a dual-pitched roof,
(ii) 2.5 metres in the case of a building, enclosure or container within 2 metres of the boundary of the curtilage of the dwellinghouse, or
(iii) 3 metres in any other case;
Where the ground is not uniform (i.e it’s on a slope) the measurement is taken from the highest part of the surface of the ground adjacent to it.
(There are other exclusions too).
You’ve said it’s 4.5m but it isn’t clear where that has been measured from, and it also isn’t clear if the building has a dual pitched roof.
If you’re in doubt you can report it to your planning enforcement team with photos and your measurements and they will investigate, but bear in mind that it might take a while. Our enforcement team are regularly sending out enforcement notices in the nick of time that they’ve had for years.
Also bear in mind that even if it is a breach of planning control, enforcement action should only be taken where it’s expedient to do so. Planning officers can (and do) use their discretion to decide against taking enforcement action.
If I was your neighbour I’d apply for a lawful development certificate before doing any further work which will clarify for them whether or not the LA consider it to be PD. It might be worth advising them of this to get clarity on the matter sooner rather than later.