Can we sort this or will we have to rip it up?

[FairfaxAikman]

Next door neighbours garden is a disgrace - he cuts the grass once all summer and it's filled with weeds.
At one point I was close to going round while he was out and pulling up the thistles as they were coming through our fence.
However the biggest problem is dandelions etc - I'm fighting a loosing battle and half our lawn now looks like the picture (scuse long grass as our mower is being fixed)

How can I fix this? If we ripped it up and started again it would be ruined again next year!

So frustrating! Would a weed and feed be too destructive on that much?
Could you dig a little order along the fence to fill with weed blocker and gravel so the creepers from next door can’t grow into the whole lawn?

The picture with the dance is the garden on the other side from lazy neighbour.

If budget allows i think you should try and add your own solid fence in front of theirs then a line of small paving tiles along that then your lawn.
I would imagine then that hardly any weeds would get through

Do you have room to have something other than lawn adjacent to the fence?

Dandelions will blow in on seed so a flowerbed would soon fill up with them too, although they would be easier to weed out. A solid fence might help. Don't bother ripping up and reseeding it will just happen again, and I would imagine turf would get infested in the end.

I am no fan of chemicals in the garden but can't see any alternative to Weed n Feed if you want grass.

Or possibly, strip, really get rid of all those little dandelions, and returf with a strip of wildflower turf? then leave that long most of the year (it needs cutting back a couple of times later on). It might help crowd out the dandelions and would disguise incoming dandelions a bit.

Thre gardens not that wide as it is so I'm reluctant to reduce the ground space as it is as DS needs a place to play.
Think weed seed and feed may be our only option but I wasn't sure if it would work

I wouldn’t use chemicals on my child’s play area just to make it look better.

I would go with @PurpleWithRed and do a wildflower patch. It would a great learning experience for your kids, and an invaluable source of food for bees and other insects.