Some advice:
You’ll be travelling during British (and French) school holidays so you should really book the Eurostar ASAP. There’s a passholder quota on Eurostar trains (unique in Europe) and some trains are already sold out. The Eurostar works like an airline and is unlike any other European train!
At what time are you landing in London? You should allow a few hours in case of delay, I wouldn’t book a Eurostar before midday. It is your problem if the flight is late and you miss the Eurostar.
On 10th April these departures are available out of London: 06:01, 16:31, 17:31, 18:01, 19:01, 20:01. Seat reservations cost 30€ pp in 2nd class. More options throughout the day with a change in Lille (see below, opposite direction).
(Since you’ll be at an airport anyway, I might suggest booking another flight from Heathrow/Gatwick. Still plan a few hours for the self-connection as you’ll have to clear immigration, pick up your luggage and do the same process again. Or fly straight to Paris from the US but I guess that’s too late.)
On 13th April all Paris-London departures are sold out (again due to the school holidays). Zero availability out of Paris on 12, 13 and 14th April. However there is an easy alternative:
- TGV Paris-Nord - Lille-Flandres or Lille-Europe 10-20€ pp
- 30 min margin + 30 min for passport controls, check-in
- Eurostar Lille-Europe - London St. Pancras 30€ pp. Departures at 09:30, 11:35, 12:35, 13:35, 17:35, 18:35, 19:30, 21:33.
So my advice is as follows: book the Eurostar reservations ASAP to avoid disappointment.
London-Edinburgh will be much easier. You can literally book a free reservation 5 min before departure (LNER website for this route, otherwise GWR).