Under birch trees, although within root distance of a larch tree in arboretum.

Looking through both NZfungi2 and combining with Mushrooms (Phillips 2006) which has UK based fungi. Would appear to be Inocybe sindonia. Going through the description:

Cap 2-4cm across bell shaped with prominent umbo. - Correct
pale buff and broken into coarse fibrous scales - correct
paler at margin - is on the younger / small one (out of focus in photograph that shows this)
Stem 40-70x5-10mm Correct using my eyeball
Slightly swollen at base - correct
White,- correct
flushed flesh colour - not sure what this means
pruinose throughout - unsure but photographs do seem to indicate this is a possibility
Flesh white, unchangine taste mild - correct
smell earthy - I couldn't place the aroma. It wasn't fungi like, wasn't radish like, not unpleasant. Not sure I would call earthy though
Spores 8.5-10x4-5 microns (spores not measured, spore shot is from washed off spores)
Almond shaped - Correct
Smooth - Correct
Spore print walnut brown (still being done)
Cystidia thick walled, elongate-cusoid with apical encrustation. Correct
Pale yellowish - cystidia not observed in real color, only with stain
Habitat with conifers. Maybe, I also note in NZfungi2 poplar, oak, willow, eucalypts are all mentioned as hosts as well. So a wider host range in NZ.
Autumn - correct
Frequent - Unsure of how to guage this wrt NZ conditions

So I would conclude there is strong enough overlap and not major / obvious reasons not to call it I sindonia.