I've ordered some of the john lewis pillows, they look to fit the bill, thanks.
My daughter is triggered by animal hair, feathers, down, pollen among other things.
She's not allergic, it just triggers streaming swollen eyes and runny nose.
The bit you linked to was specific to dust mite allergies, and, as it says further up:
Many people with pet allergy do not need any tests as the cause of the allergy is clear. Symptoms are worse in the home where the pet is. Symptoms improve when out of the home, especially for longer periods, such as on holiday.