Yes, for much of the first half of my life Ireland was my home.
I do not think we ever established how long you are going to be in Europe, but for the trip you are thinking of IMO two full weeks would be the minimum and more would be better. With two weeks I would give five full days each to London and Paris and three days to Nice. You should try to arrange your travel so you fly to London and home from Nice.
Book your train travel ahead for the lowest fares. Use http://www.voyages-sncf.com or www.tgv-europe-com for the trip from Paris to Nice. I think you will have to book the London to Paris trip on http://www.eurostar.com using the UK as your "home" country. When using any of these booking sites do not select the US as your home country or you will be diverted to the RailEurope site where you will pay twice as much for your travel with fewer options. Go to http://www.seat61.com for a full run-down on how to navigate the booking process as it is not always self-evident.
Eurostar tickets can be booked up to 120 days ahead IIRC, iDTGV tickets to Nice up to 180 days ahead and regular TGV tickets three months ahead. The earlier you book the better your chance of getting the lowest fares, roughtly £30 for the Eurostar and as low as €19 for the Paris to Nice bit.
As a general rule trans-Atlantic airfares drop significantly around mid-September when they switch from Peak to Shoulder season. I would hold off at least a few months before booking. Check the fares on e.g. Kayak, Mobissimo and Bing and any other booking sites you fancy. If you check every few days you should be able to see if there is any trend.
One thing I would mention is that September is a VERY heavy month for hotel bookings in Paris because of trade shows and conventions so you might want to start looking for a Paris hotel sooner rather than later and in any case before you book your flights.
Have fun planning!