I'm sorry to hear about your situation. Clearly, both your parents need support, and that must be very worrying for you.
You probably know this already, but as you mention your dad has cancer, I wonder about contacting the MacMillan Nurses for support for him? I hear very good reports.
Info is here:
http://www.macmillan.org.uk/information-and-support/coping/getting-support/macmillan-nurses
They have a support line you can call, Mon-Fri from 9 am to 8 pm at 0808 808 00 00
It could not hurt to give them a call and see what they suggest.
I agree with arranging for an appointment with the memory clinic (a home visit perhaps), and not asking your mum if it's okay, and also not telling her about it in advance. Just arrange it. If she objects when they turn up, lie and say it's a visit for your dad. Or if you have to take her, say you're going out for lunch/tea/a drive/an appointment for your dad, and just take her. Don't tell her ahead of time, and don't let your dad talk to her about it in advance, either.
I don't mean to upset you, but there is no point in trying to have a logical discussion with someone with dementia, or to get them to "accept" their diagnosis. Reason and dementia don't mix and if you try to reason, you will likely only provoke upset and distress, especially if she tends to be anxious anyway.
A lot of PWDs (persons with dementia) have something called anosognosia, which is the fancy medical term for the inability to understand/comprehend there is anything wrong with them, due to the damage the dementia has caused their brain. They just can't grasp that there is something wrong, so there is no point in trying. It's like asking someone with a fractured leg to weight bear on it.
So bear in mind it may not be denial in the classic sense of your mother refusing to listen to reason or accept something. That may be true, or partly true, but it may be that she just cannot.
For what it's worth, I have been there and done that with own mother, before her diagnosis, and it was incredibly difficult and frustrating and awful, and you have my sympathy. But I didn't have another ill parent in the mix, as you do. As you say, for your father's sake, you cannot afford to wait for a crisis.
So I wouldn't suggest anything to your mum, or give her any choice, I'd just get on with it. By all means have conversation with your dad, when he is up for it, and privately, away from your mother. Get your dad to give permission for you to talk to the GP about his care, and then address your concerns with the GP. And remember that even if the GP can't discuss something with you without permission, they can always listen to what you have to say. You can document everything that happens and then provide that to the GP and also have it for Social Services.
Others here will know more about the UK system but at a minimum, I think your dad is entitled to a carers assessment. I'm not sure where you start but perhaps with social services? Others here can point you in the right direction.
And I hate to give you more tasks for your to-do list, but if you don't have power of attorney (both for health and welfare and finances) sorted for both your parents, I'd get on that as well. I fear you will need it sooner rather than later.
Again, I'm sorry to hear about your situation, and hope you are able to get some help and support sorted out. Best wishes to you and your family.