Legally, it will all come to you, but morally (as you say), none of it should really belong to you, as your connection had long since ceased (assuming your children are now grown up and are no longer dependent on you financially or for housing, feeding, care etc.).
Anything that comes your way should morally be passed straight on to the children. That said, he obviously had a relationship (albeit unmarried) with his new partner, so maybe some of it should be offered to her. If they had no children, she presumably wasn't financially dependent on him as a SAHM.
It might be a weird way of looking at it, but he had a significantly lengthy connection/relationship with both of you at one stage - as he had no children with his new partner, should it make a big difference that his relationship with her was the later one, which happened to be during which he died? Is that a really crazy thing to ask?! I suppose it also depends on the circumstances of your separation and if there was acrimony or just growing apart.
All else being equal, I'd say that, morally, all liquid assets should go straight to the children. If they had a jointly owned house, which was bought after they'd got together, then they will either have bought it as joint tenants, in which case it will now all have passed to her, or as tenants in common, which will have been his way of saying (legally) that he didn't want her to inherit his half on his death. Possessions and chattels, obviously, are likely more of sentimental than financial value so the children will probably need to discuss that with the new partner.
Just one thing for your children to establish, though: will she be planning to leave everything (or at least your former husband's share) to your children on her death?
Sadly, it's all too common in such cases for the surviving partner to accept everything as their own (in her scenario, more likely chattels and sentimental positions/heirlooms already in the house that won't be accounted for in any will or considered for probate) and then completely ignore the connection with their late spouse/partner's children and give them nothing, automatically passing the lot on to their own children (if applicable) or families/friends.
It's not always done consciously, but it frequently ends up that way when the 'link' person dies. When my (widowed) GM died, and my DM (her DD) died soon after, almost all of her possessions, including all of the sentimental/keepsake things, went to her other DD, my DA - made perfect sense as she had been her DM. My DA has now died and all of those things of her DM's which previously went to her and had since become absorbed into her own worldly goods were dealt with, as you'd expect, by her own children, and all passed on to them. Through nobody's conscious planning or deliberate conspiring, my DSis and I (and our DC) will never have a chance to own any of those things and their many associated happy memories - at least not without seeming very grabby or things getting very awkward, considering we'd now be effectively asking our cousins for their Mother's possessions.