Your post really touched me OP, I've been exactly where you are. My dad was diagnosed with FTD, a rare and aggressive behavioural variant of dementia aged 60 and it was truly heartbreaking to watch a man that was so incredibly proud and strong with a love for life struggle to battle such a cruel condition. It's unlike any other disease and whilst your dad has the diagnosis, it's something that you're all living with and learning about together.
From experience, I know it can be really difficult to access support because your dad is under 65 and there's just not the same availability than if he were just that bit older. Even when we were able to get help, it often wasn't relevant because of the stage in life he was at - caring for an elderly parent with dementia isn't the same as caring for a young, fit parent or partner in their 60s so I'm not sure how much value you'll find looking at the elderly care resources.
We found that doctors weren't a particularly great help because they didn't understand the condition so were essentially left to our own devices, however, Alzheimer Scotland were fantastic and we couldn't have made it through his illness without them. They told us exactly what help, support and services we were entitled to and how to go about getting them as well as how to help manage my dad in difficult situations. They also offered counselling and a support group for "young carers", again because caring for someone so young with dementia is a beast of its own kind. If you're in England then I imagine that similar charities will offer the same help and I'd definitely recommend exploring that.
As for your dad being upset with the neighbours, we went through that a lot. Distraction never worked with my dad - once he had a bee in his bonnet there was no getting him to move on. Yes, he'd take a biscuit and cup of tea but then he'd be right back at the thing bothering him before. As others have suggested, we found that acknowledging his feelings was the best way to go "that's really awful, they're just mistaken and I'm sure they'll realise that, we'll speak to them tomorrow and sort it out". It might take some time to convince him but it certainly worked better than telling him straight he was wrong. If anything it only riled him more.
If you know and trust your mum and dad's neighbours, it may be worth mentioning his diagnosis to them so they can understand any strange behaviours and can keep an eye out for him, especially as his condition progresses. So many people misunderstand the condition and think it's all about forgetfulness which of course it's not. We found people to be particularly unforgiving of strange behaviours because my dad was so young and it just wasn't expected that he'd have dementia to justify the things he was doing. Once you're comfortable sharing the diagnosis and the difficulties you're facing, it really is a weight lifted and can help manage the condition and challenges it brings.
My last thought is that dementia is incredibly isolating for the carers and both you and your mum need to look after yourselves so that you can look after your dad when he needs you. Make sure you both take time for yourself to unwind and recharge, however that may be. Wishing you well OP.