hello
@Lily may
and welcome to TP
a referral to your local memory clinic comes through the GP
as you're not entirely sure what was the outcome of the last visit your mum had to her GP, it may be worth writing out a list of all the changes in your mum's behaviour and her ability to do everyday tasks (think of her 1,2,3 years ago and compare) together with all your concerns, then send this to her GP - the GP may not be able to discuss your mum's health with you, due to patient confidentiality, but any info you give them will be noted on her records - maybe keep a diary over a couple of weeks so that you have specific and recent examples
you might ask the GP to invite your mum in for a 'well woman' appointment; could you go with her so that you hear what is actually said, and can indicate to the GP when your mum is not quite giving the whole picture - or if your mum is happy to visit the GP, suggest to her she go for some reason she will accept eg a review of current medication
if your mum is happy to have you support her, she can write a letter to her GP saying she gives her permission for you to be involved in her care and for the GP to talk with you about her health and be involved in making appointments and be present at consultations