My guess is that this pipe is not connected to the main drains at all.
Have you monitored what's coming out of the end?
It's a combined sewer system that you and your neighbours have, that means all foul water and surface/storm/rain water all goes down the same system. If there was a blockage/leakage/overflow into your garden the smell would be horriffic by now and you'd have a deluge. It would be overflowing down the embankment.
Dare I say it, but this plan is a red herring.
Well, it's a bit confusing 'cos I suspect that the real drains are deeper and larger. (And why an inspection chamber was not incorporated at the four way junction in your garden, only the installer knows).This isn't the kind of pipe that would have been used in the system.
I suspect that it was an attempt at a soakaway, possibly from your patio or similar that is now useless.
Also, I suspect that 'cos of the lie of the land all your neighbours garden rainfall is finding its way to the bottom of your garden.
Now if they had done something to collect/divert the rain to your garden you could investigate the legal options. But, 'cos I suspect it's just the lie of the land that means its your garden that's the lowest, and your neighbours haven't done anything, well, that's the way the cookie crumbles sometimes. (Although they might have raised their garden levels which could be construed as a nuisance, but only if it was lower than yours to start with.)
I suggest that you follow this pipe and either lift it out, or, the other end may give you a clue to its purpose, if it has one, in which case it is, as I suspect, a defunct soakaway, then reinstate the soakaway.
After identifying the purpose and reinstating the soakaway, or removing the pipe completely, then monitor the situation.
I suspect that it won't make an iota of difference. In which case you're faced with a different kind of problem that will need a different approach, more along the lines of landscaping.
But if you're really convinced that the drain is blocked/broken then I suspect it really is a professional that you need to reassure you or identify the problem, because of: a) the scarcity of access chambers and b) the size and depth of the real drains.