@OnTheFencePaint Doesn't have to be the exact same street though. One of my posties technically lives two streets away, but the way the streets are laid out, the back entrance to her house is probably less than fifty yards from mine (I’m nearly on a corner). I often see her walking her dog within minutes of delivering my post, so obviously she walks her route back towards her own house, with me being one of her last deliveries, and then clocks off. She also happens to have two DC at my DS's school, one in his class. Fortunately we get on! But there's no way to legislate against a postie knowing some of their 'customers' outside of work.
If I was the postie in the OP situation, and really felt awkward about it, I'd probably have a word with my manager and see if there was any way to get moved to a different patch. But it may be that it suits them because they want to finish their round close to home, like my neighbour. Or are they maybe less bothered about the situation than the OP is?
Falling out with neighbours can quickly get very toxic and evolve into decades-long feuds, even over something very minor. Honestly my instinct would be to try and nip it in the bud. Maybe next time you sign for a parcel, would it be worth taking a deep breath and suggesting with a smile that the two of you bury the hatchet, now that you're going to have to keep coming face to face like this?
Obviously if the dispute is over a practical issue which you actually need resolved, then it's slightly more complicated than that, but still, taking the leap to try and find a compromise might be worth it? Your neighbour may well be very relieved - it can't be fun for them either, going to work every day not knowing whether they're going to have to face you and this awkward situation. They might be terrified that you're going to do or say something which might jeopardise their job (I'm not saying you would of course! But they might think it.) Either way, it best not to let it fester.