Hi @Glitter99x. I am sorry it all went pear shaped for your DP's grandmother.
A couple of things though. Most times a GP will refer a patient to a memory clinic - usually attached to a local hospital. My mum was suspected with dementia (Alzheimer's and Vascular) in October 2013 and didn't get a memory clinic appointment until December 2013. We got her diagnosis in February 2014. These things do take time. My mum had to have a pair of clinical nurses from the memory clinic do some basic tests (like who was prime minister, what day it was, what her writing was like, her cognitive responses), then they recommended the memory clinic for CT scan etc.,
You can (or your DP's mum) can go and see a GP to discuss this or telephone them with your concerns. It could be that the GP could refer your DP's grandmother direct to the memory clinic (if there is one in your area). It's unlikely the GP could make a diagnosis without referral to a memory clinic (though I stand to be corrected - that's how it happens in this area). Some memory clinics do need GPs to do certain tests ahead of the appointment.
As far as medications are concerned it depends on other medications. For example, my mum was on warfarin and insulin. She was not offered dementia drugs because they can make users dizzy. Having a dizzy spell could result in a fall which could cause diabetic coma or bleeding on warfarin. As she lived alone it was felt it was too dangerous.
The issues around urgency are things like: getting power of attorney set up. If the person has finances/property someone has to be power of attorney. Banks can't talk to anyone, not even spouses, without it and that can cause hardship.
Personally I would call the GP (your DP's mum is the best option or his grandfather as they are next of kin) to discuss the issue.
The other people who may be able to help is the Admiral Nurse organisation. They can advise on everything around dementia and are there to support families. They can help with how to spot the signs and may be able to advise you on how to get the diagnosis. The normal 10 minute appointment with a GPis probably not long enough. They'll know the procedure in your area. They can't pull strings obviously, but they can advise you if there's a structure to follow to save you wasting time getting appointments with the wrong person etc., Their main number is 0800 888 6678
Good luck. My mum is now in a nursing home - we managed to keep her at home between us and carers and GP surgery staff for 4 years before a series of falls made it too dangerous. She loves her new home, is always smiling and we know she's safe so that's all that matters.
Do get someone to make that phone call to either the Admiral Nurses or the GP to discuss this. It takes time to get into the system and whilst three weeks may just be what has to be endured, it'd be good to know that is the case rather than finding out a simple call could get her into the system quicker.
And do make sure your DP's grandfather takes every bit of help available to him, from carers allowances/attendance allowance, to day centres (to give him a break), and carers in the home. It is his and her entitlement, it is not charity, begging or a case of he can manage. My GP told me that his job was to keep me well as I was main carer for my mum and my DS - as he said without me everyone falls apart. Without your DP's grandfather everything falls apart. Keep him well and give him support.