OP, I once read that people who were happily married tend to move on more quickly than people who weren't.
My ex left me to come out as gay. I know it is not the same, but it felt very much like a bereavement. The man I thought I was married to 'vanished' almost overnight and I felt bereft.
I met someone else very shortly afterwards. We were both recently separated and of the few people who who knew about us, more than one said it was too soon, we were just propping each other up in a difficult time, etc. I felt so much shame/guilt at falling for someone so quickly that I broke it off. What would people think of me? (OK, it was a bit more complicated than that, but that was a big part of it. My ex wasn't out yet, I thought people would think I'd had an affair).
Anyway, the new man and I kept in touch and we got back together about 18 months ago. We are really good together, but integrating our lives very slowly. Not least because both our exes introduced new partners to our respective children very quickly and neither of us wants to burden them with more change, too fast. When we first met, it probably was too soon for both of us, but that doesn't mean that the relationship was/is the wrong one. I can see why people think you were vulnerable, but perhaps your partner is just a very caring person who felt a connection and wanted to support you. Relationships are so much more nuanced and complex than a post can convey.
Your mum is just wrong, in my opinion. And quite cruel, really.
I don't think it's a good idea to keep the different areas of your life completely separate, but that doesn't mean you have to merge them fully either. It could also be positive for your children to see that you can find happiness again after a loss. I also think it's been helpful for my children to meet my new partner occasionally. They know who it is I spend time with when they're not around, so they don't worry about me being alone or who I'm with, but they also know that my time with them is my time with them and there is only a little overlap between these two parts of my life. I hope to increase the overlap a bit over time, but definitely no plans to blend.
Good luck with finding the right balance for you, your partner, and your children.