blue…. You were right to post the question which is bothering you, rather than a general ramble about grief, your children etc., though as you can see, some have judged you for this. I'd ignore those comments.
My wife died a few years ago, and I have children of similar ages to yours. There is no time limit for starting a new relationship, but in my view the important thing is that, before you try, you are confident you have come to terms with your loss and started to live in your new normal, instead of being stuck in the emotional bomb crater which is the lot of the recently bereaved and looking for someone either to pull you out or join you in it. This can take months, it can take years. Only you can make that decision. The downside of starting a new relationship too soon is that if it fails - which is quite likely - you will be back at square one with unresolved grief, made even worse by the loss of the new partner.
It's possible the feelings you refer to are not so much romantic as the joy of being cared about. The loss of this is one of the hardest blows the death of a spouse inflicts. My suggestion would be to proceed gently and see if this woman continues to involve herself in your life after you have ceased needing the help she now offers - 6 months was when the extra assistance I’d had in the early days started to drift away. See how things develop over the next few months. If she sticks around, choose your moment and ask the question. Even if the answer is not what you hoped, I’d guess she will understand, and the friendship can continue.
Should matters progress, I’m sure you will involve your children and consider their feelings, but it’s very possible their reaction will be ‘how great to see Dad happy again’. Friends…. who cares. It’s your life.
So ‘Women mourn, men replace’? 'Replace' is rather insulting. Widowed men go on to new relationships simply because they can (though I am still on my own). As far as many fifty-something unpartnered women are concerned, so long as he has dealt properly with his grief, a man widowed after 26 years of happy marriage is a rather better bet than an embittered, penniless divorcee or a never married, un-housetrained commitment phobe.