The other person nailed it - time is basically everything here. Your nan wasn't doing anything fancy, she was just letting a whole chicken (or a good pile of bones with some meat attached) just hang out in water for literally hours. That's where the magic happens. The collagen in the bones breaks down and turns into gelatin, which gives you that silky, rich mouthfeel that makes it actually taste like *something*, not just hot water with chicken bits in it.
Here's what I'd do: throw a whole chicken or a bunch of bones/carcass in a big pot with cold water, bring it to a boil, then immediately turn it down to a gentle simmer. Skim off the gross foam that rises to the top in the first few minutes - that's impurities and it makes a difference. Then just... leave it alone for like 2-3 hours minimum. Throw in some carrots, celery, onion (cut in half, doesn't need to be fancy), maybe some garlic, a few peppercorns, salt. Don't go crazy with seasonings because you want to taste the actual chicken. If your nan was old school she might've added parsnip or parsley too. The long simmer does all the flavor work for you.
The trick most people miss is that they're impatient or they use chicken breasts instead of a whole bird. Breasts are too lean - you need bones and skin and the whole situation to get that deep flavor. Also strain it properly when you're done and honestly, let it cool in the fridge overnight if you can. The fat congeals on top and you can see exactly what you've made. That's the real deal. Your nan probably made it the night before too, which is why it tasted so good when she heated it up again.