Layover

no it's cutting it too fine

any delays and you will have to buy a new walk up ticket from London to the US

It will take you an hour to get off the plane and through passport control then check in times for flights to the USA is 3 hours due to USA government checks, so no, your connection is too tight. Even if you could use flight connections rather than going through passport control, you wouldn't have the necessary 3 hours. And that doesn't allow for any potential flight delays. Sorry, you need to allow longer.

Thank you for your help. Do you think the 5 hour layover time on the way there would be fine? (Same situation)

Also.... Lufthansa or Austrian Airlines?

5 hours should be OK on the way there, because London to Geneva will only have a 1-2 hour check-in.
But, some questions:
Q1) Which airport in London? - Heathrow is worse than Gatwick or Stansted
Q2) Which terminals? - changing terminals at Heathrow adds another 30-60 minutes.
Q3) Will you have luggage? If you do, you have to go through UK immigration, pick up your luggage and check in again - add 30-60 minutes. If you are hand luggage only, and have already printed out your boarding card before you left home then you can shortcut as an "airside" transfer and don't have to go through immigration.

That said, it could still go pear-shaped if the first flight is late of immigration backs up. But buying a new ticket for a later flight to Geneva is probably an acceptable risk.

Overnight in London. Spend a relaxing afternoon and evening. Take the next day's flight. Stress free.

Have you looked at flying to Zurich with Icelandair? It is $500 cheaper than flying to Geneva. You will have an additional 3 hour rail connection for about $100.
Where are you going in Switzerland, or is it France?

"Also.... Lufthansa or Austrian Airlines?" errr, neither. Where are you seeing these flights? At best these are Star Alliance codeshares on Swiss (who, like Austrian are owned by Lufthansa). At worse you are going via Frankfurt or even more off track with yet another change of plane.

Be very careful that all flights use the same London airport, the travel time between Heathrow and Gatwick or City will be up to 2 hours, to Luton or Stansted will take even longer (on top of your immigration and baggage waits and checkin for the second flight)

5 hours to change airlines at either London airport is a good cushion if your flight from Anchorage arrives on time. Anchorage in December -- are flights often delayed or cancelled because of weather? That's something you need to find out. Look up the historical on-time percentage for your flight to Heathrow or Gatwick.

Your return is NOT good. 3 hours may not be enough time especially if you're changing terminals. Flights also close 30 minutes before departure so your time to catch the flight to Anchorage is reduced to 2 1/2 hours. If your Geneva flight is late, you're in trouble.

Are you prepared financially to buy a walk-up one-way ticket back to Anchorage on the next flight in which there is a seat available?

Thank you all so much for your help! I found tickets that fly from London to Geneva and leave at 1:30pm, now giving us 6 hours. We fly into and out of the same terminal at Heathrow. Yes, with luggage.

On the way back with only the 3 hour layover we will relax and spend the night in London.

We are staying a few nights in Geneva and then heading to Chamonix for a week.

Is it $500 per person or $500 total?

David's suggestion to stay over in London is probably the safest choice, but adds expense. If the $500 saving you mention is per person, staying over makes sense.
If its $500 total, you may be better off spending the extra $500 on the air fare to have a one ticket itinerary and let the airline determine your transfer time. At least that way your checked baggage would go all the way through to your destination, without your having to retrieve it to change planes.

Three hours on separate tickets/itineraries, has worse odds of success than the long term survival of the proverbial snowball in the devils domain.

$500 per person.. Our schedule is going to be the same so I don't think it matters where we stay? We are planning on London so we don't have to stress about the short layover.