"It’s a low fence, only about chest height, and it’s their fence so we can’t do much about it. But I’ll look at things like a trellis."
You can put up a higher fence of your own inside your garden. You'll lose, what, maybe four to six inches of your garden? Worth it for the privacy, surely? I would mention it to them before doing it, just say you'd really like more privacy so that's what you're doing.
"The new neighbours are clearly very pally with the neighbours on the other side of them, they go over to each other’s a lot and I think I’m just sensitive at finding myself in another situation where I feel ganged up on because they’re all friends and don’t want anyone else in their patch."
You're making assumptions based on your old neighbours, not these people. They get on well with their other neighbours, what's to stop them getting on well with you? Nothing, except how you choose to behave towards them. You say you "feel ganged up on" - but is that feeling just sparked by memories of your old neighbours? It sounds as if your new neighbours haven't pressed themselves on you - they're giving you the 'social space' to be as friendly or keep-to-yourself as you wish. That's not 'ganging up' on you. Quite the opposite.
"FYI our previous neighbours were constantly outside having people around for barbecues and drinking and that’s probably what’s putting me on edge with summer coming and thinking they’ll be the same."
Can you see that you being on edge is all about your old neighbours and not about the new? You're making assumptions. You could be wrong.
"We probably didn’t get off to the best start when we moved in, I didn’t mention that there’s stuff like we share a front path that leads into a shared side return and it just feels like they’ve claimed it as their own with stuff like painting the front gate and putting up lights."
When did this happen? In the three months since you moved there - or before? If before, how do you know it wasn't with the agreement of the previous owner of your house? If since, did the gate need painting? Are the lights security / motion sensor, or decorative? I don't see it as 'claiming it as their own'. Again, I think you're not seeing these new neighbours, you're still dreading the old.
"We probably didn’t get off to the best start when we moved in"
What does this mean? There's a big difference between you just looking at the gate and the lights and feeling a sense of dread but saying nothing, and you actually saying to them they shouldn't be doing that. What do you mean?
"I just wanted things to be normal and peaceful after feeling on edge all the time."
In my opinion, there is no evidence in what you've written that things cannot be normal and peaceful. You DO need to deal with your feelings. It seems to me that you are still feeling on edge all the time, and that is not good for you. I think you should take a deep breath, invite them round for tea/coffee or drinks, and get to know them. See that they are NOT your old neighbours, these are totally different people. Settle your fears that it's all going to be like the past.
It would also be a good time to bring up your plans for increasing the privacy in your garden. Maybe discuss the shared path/side return. Not in an accusatory way, but along the lines of 'what has happened before about shared maintenance of the area, how often does the gate need painting' kind of way. Get to know them. And settle your fears. ((hug))