I wouldnt try for the Embankment. It's a massive crush. Also, by about 11.00, the Police close the footpaths leading down to the Embankment, because it wouldn't be safe to let any more people down.
I'd suggest Trafalgar Square. It gets busy, but it's so big that it's not squashed. The atmosphere there is really lovely ... everyone talks and laughs with each other ... possibly the only time that ever happens in London.
There are loads of families there. You can see the fireworks looking down Towards the river, over the tops of the buildings. They have a giant screen with a countdown clock and, up until midnight, they show music videos on the screen.
And because there are so many exits from Trafalgar Square, you can get out quickly and comfortably afterwards.
You don't get as good a view of the fireworks as you would standing right on the river, but it's a lot easier, especially with children,
And the restaurants around the Square all stay open so you can be having a pizza or a burger until 11.00, and then walk along to the Square itself.
The Square is closed to traffic, so you can walk out along the roads. They even close off Regent Street, which is eerie ... although you'd be going the other way if you're staying near St Paul's.