oomis
I’m not a posek, but I would imagine that the practice of not cutting nails in order has more to do with the one getting them cut and not the manicurist.
Women going to the mikvah do try to cut one set of nails on the previous day. For those who don’t cut nails on Thursdays in general, they cut the day prior. However, if one forgets altogether, then better to cut the same day, although it’s not optimal.
As an aside, one should remember to wash netilas yadayim after cutting nails (or going to the barber)
As always, consult your LOR