The mould appears in the area behind and adjacent
to some lined curtains both on an externally facing wall

Condensation occurs where a surface is below the dew point, curtains
are insulatory to some extent and may result in an area of wall being
cooler thus local condensation and mould.

and the adjacent internal wall.

That suggests higher humidity than desireable from somewhere.


Can you verify no sources of moisture penetration?
- gutter leak, radiator leak
- sink/shower/bathroom leak, poor sealing thereof
- leak around window frame (wind blown)
- leak around window-cill
- damaged render or blocked vent(s)

Can you be certain occupants ventilate it properly?
- kitchen & shower/bathroom extractor *actually used*
- trickle vents not stuffed up *when you are gone*

If all ok then frankly I would spend the £100 on CWI.
CWI will make the external walls warmer and they will check the cavity
for you for free - perhaps get them to check right where you have the
condensation for anything bridging the cavity. CWI will not help the
condensation on the internal wall, which makes me suspect there is
inadequate ventilation.

It might be better spending £100 on getting a humidistat set on the
relevant fan(s).