On the Boots website, there are roughly the same number of 1-star reviews (8) as there are 5-star ones (7), with just two people giving the product 3 stars.
One of the 1-star reviews reads as follows:
"The brush heads are almost impossible to attach and if you do manage it, tug of war to remove them. Mostly they just fall out of the handle and so the item is useless".
Another says:
"The straight brush is easy to attach but the two round brushes are not. Instructions say push until it clicks into place. Neither did. I was reluctant to push too hard in case I damaged it. Took it back to Boots today. 2 assistants attempted to attach the brushes without success. They then tried an alternative which they had in store. Again with no success. I was refunded."
This is also shown:
"Even for the low price I would have expected better quality. The brush attachments come off and after only 3 or 4 use".
. . . along with this:
"I bought this as it has both hot and cold air button and button to release the brushes. However i had problem attaching the brushes, and once i have attached it was almost impossible to detach."
https://www.boots.com/tresemme-full-finish-airstyler-3-interchangeable-brushes-10193302
So it seems that you're not alone in experiencing difficulties in attaching and/or removing the brushes.
I suggest returning the product to Amazon for a refund.