Even if you dont want to fly with them, but the Qantas website gives you a good idea of how you can plot out your trip if using a "Round the World Fare". Its called Oneworld Planner. You can find it on their website.
You definately cant do the trip in that order using the RTWF. You could go Egypt, London, NY. Or you could go NY, London, Egypt.
London and Egypt aren't too bad either way round, just a short flight (2-3 hours I think and similar time zones) - but don't sandwich New York in the middle of them! That would play havok with your body clock.
The week before Christmas is great for touristing in London - book accomadation early and pick what you want to do, avoiding the main shopping centres (which aren't really that bad... but then I have lived in London so am used to the over-packed streets). It will really depend on the weather how things go with London and what you will want to do. The open-topped tour busses are NOT so good in the rain! Nor is the walk from the Houses of Parliament to Buckingham Palace. But some things can be done in the rain (London Dungeon, for example, is great; Tower of London is a bit exposed in places) and there are some excellent museums.
I can help with NY, we've been a few times. If you book early enough you shouldn't have any trouble finding a spot. How long will you be there for? Short-stay apartments are a good way to go if you're there for 5 or more days, especially with your DD. Mid-town is handy to everything or you could go a little bit further up to the Upper West or East sides to be near Central Park and great places to eat. NYE at Times Square would be amazing but hectic, I think you need to stay inside a cordoned-off area for hours so bear that in mind. Incredible experience though.
In terms of seeing the city, the hop-on, hop-off red bus tours are great. We also did a Harlem Gospel tour, a walking tour of Brooklyn and a dinner cruise on the Hudson. Loads of activities for kids, especially at that time of year - ice skating, all the department store windows will still be gorgeous; plus the Central Park and Bronx Zoos, carriage rides, Museum of Natural History, cupcakes at Magnolia Bakery (well, that's for all ages - OMG, yum!), a show like Lion King or Mary Poppins, trip out to Coney Island....
Subway is dead easy and so convenient.
Have fun planning, I'd love to go back! Hope you enjoy London and Egypt too.
Bookmarks