Depends on how much you hate gardening but either dig up the weeds or get a gardening firm to come and do a one-off clearance for you. Then plants lots of ground cover plants that need little attention and also keep the weeds down. Scroll down to the bottom of this page, click on G and then Ground Cover for lots of possibles.

Also extend the lawn if you think the beds are too big to manage so the only real job to do is cut the grass.

Once your neighbours see you are starting to tackle the weeds, I am sure they will be encouraging and friendly.

Have a weeding house warming party lay food and drink on and get all your friends around to help it works a treat a colleague did the same then lay some membrane down to suppress the weeds and stop them returning and look for some low maintenance perennials

Charming! If my new neighbours mentioned the weeds to me, I'd just be saying politely you'd got too much to do in the house to worry about those just at the minute, but that's just me.
If you don't want to tackle what's out there right now, nor have any intention of creating a garden yet, weedkill the lot - apply a path weedkiller such as Pathclear - should see off most things, but you won't be able to do any planting for a minimum of 3 months, preferably 6. It won't, though, deal with things like brambles or anything else woody and large.

I so agree with Bamboo, I would say the same, settling in to the house comes first and I will in my own time get around to the garden. Do not feel pressurised in to doing the garden take your time deciding what is the best way to tackle it. If you have inherited a mature neglected garden there may be some treasures hidden in the weeds or bigger shrubs/plants that need TLC. Why not photograph it and show us the problems.

welcome to GoY. the neighbours may not realise how they come a cross. what is their garden like? if they are keen gardeners ask their advice. Chose a dry day and ask them round for coffee/cuppa and be bold say you'd like to see what you have got and if anything is worth keeping. you never know they may even help you with it. During the conversation drop in about how much you have to do in the house etc.
If its overgrown then there may well be some lovely plants waiting to be 'helped'.

I think I would cut the grass and perhaps strim the borders so everything is short.

we are good at identifying plants on here too.

'Weeds'. are just flowers on the wrong place.

As we have no idea what those weeds are... Wait until summer is here, then dig them out (before they seed).

If the neighbours are that concerned, they might come and help.