How much time to clear customs/immigration?

It might help to spell out the cities you want to fly to (not everyone knows airport codes). I would allow a minimum of two hours to transfer at Amsterdam or Oslo, but 2.5 would be better in Amsterdam. I would also double check there are later flights in the day in the off chance you miss the connection. If you are returning home from London, I recommend you do the same as on your arrival (fly in the night before your return flight to Seattle). FWIW, if you were willing to change airports there are direct flights from London Gatwick to Bergen on Norwegian Airlines. You'd have to calculate the transfer time into the equation but you may save yourself a lot of flying time.

You will need to get an ETA to leave Heathrow and enter the UK. There are MANY posts in the forum about how to do that. Only get one from the official website and be careful of third party sellers that look official but charge twice as much.

if you're flying from Heathrow to Bergen via Amsterdam it'll be with KLM so leave it to the airline to decide how much time you need - it won't be much. You can do transfers at Schipol in about an hour.

I like the idea of flying direct whenever possible and if within budget. If you opt for the direct flight from London Gatwick, I would consider spending the night closer to that airport depending on your departure time.

If you opt for the connection from Heathrow, I would be inclined to go through Oslo. I think there would be more options for flights to Bergen if you were to miss your connection. I think Scandinavia Airlines has several flights a day for that segment.

I loved Bergen! I also did something similar when I went there. I flew Delta from US to Copenhagen (spent few days there). I bought a separate one-way ticket on Scandinavian Airlines directly to Bergen. Bergen is a small airport and I only had positive experiences in Scandinavian countries overall. This was before EES, but I didn't have checked luggage and I had mobile boarding pass. I just scanned it at Copenhagen airport to enter the security/x-ray screening area (nobody checked my ticket or ID!). After security, I went to the gate. To board plane, I scanned my mobile ticket at a turnstile while the agent was at the desk away from it.

The same happened when I left Bergen to Stockholm. No ID checked when entering security/xray, and scanned mobile ticket at one of turnstiles to enter jetway. I joked that it could be someone else traveling with my ticket as my ID was never checked. Only checked when I left Stockholm back to US.

You'll go through immigration when you enter the UK.

You'll go through immigration when you enter the Schengen; in scenario one, that is AMS, in scenario two, that is Oslo.

You are correct. I missspoke. My bad. Still 2 hours in either is my preference especially now with the fingerprint nonsense. If the airline wants to sell me a ticket with a 1 hour change, I wont take it. 90 minutes, maybe if its a lot cheaper or gets me in at a better time. But not my preference and I dont have to do the finger print thing (maybe).

If it were me, I would definitely get the direct flight from Gatwick. It will be much quicker than connecting thru Amsterdam or Oslo. Once you arrive at Heathrow, there's a direct bus you can get to Gatwick, so you can spend the night at one of the airport hotels there and fly the next day. The flight is less than two hours.

That new Alaska nonstop to LHR is a game changer for Seattle passengers. Here are the price comparisons for random day in June
Alaska non stop, 30k points +$6
American Airlines 1 stop, 70k points +$19
British Airways, 1 stop, 45k point +$567

You can understand why the OP chose to take the Alaska Flight to LHR and then connect separately to Oslo.

One other option to consider- Seattle to Helsinki costs 35k points +$38. You take the same Alaska flight to LHR and transfer to FinnAir flight to Helsinki. I haven’t checked connections between HEL and Oslo and compared price wise for your separate flight, but you may want to compare your options. Remember flights on One World Airlines still earn some points. Status only if frequent flyer points used, but regular points and status points for purchased flights.

Carol now retired… yes, it’s wonderful to have this new option for flights! We bought that flight as soon as we decided to go, so maybe 3+ weeks ago. Helsinki is not part of our plans, this trip anyway. I did book an SAS flight this afternoon LHR to BGO via Oslo. Going to cross my fingers as the connection is 1 hr 20 minutes.
We spend 3 weeks total between Norway, Denmark and Sweden, and we fly back from Copenhagen on Delta. We’re pretty excited!

Terry OP - Have a fantastic trip. Scandinavia is truly a beautiful region. Glad you got your tickets and took advantage of that special new flight.