We are Americans flying from Chicago (O'Hare) to London (Heathrow) and staying for 5 days. Will we need to go through both immigration and passport control? Customs? How long should we plan between arrival and getting into a cab?
From there, we are flying from London City Airport to Zurich and then taking the train to Adelboden. Will we need to go through passport control or customs when we land in Zurich? How long should we plan between arrival and getting on the train?
I think you can count on having your passport checked at every step along the way. You know that it will be checked against your boarding pass at every security point to board a plane. Immigration and passport control are the same thing when you land. They will check your passport, ask a question or two (how long are you staying? etc) and stamp it. That will be both landing in London and Switzerland. Customs is merely exiting through the green (nothing to declare) door after you pick up your bags.
Personal experience with several entries at Zurich airport, immigration was quick and painless, usually at the luggage carousel before the bags came out. There will be a connection to Adelboden every hour, from the train station in the basement of the building across the street. The last leg will be by postbus.
Your Heathrow Immigration experience can be anywhere from 15 minutes to over an hour, depending on which terminal you arrive at (separate immigration points for 1 and 3, 2, and 5), how many planes have arrived around then,, and again how far you have to walk (our last trip was an American flight which uses the most outlying T3 gate, that was a walk of over 10 minutes to Immigration, it was about an hour total before we got to the Undergound station.
Upon arriving in London, you will go through passport control, which in this case is the same thing as immigration. The wait can vary tremendously, depending on how many agents are on duty and how many other flights are arriving. Heathrow is one of the world's busiest airports and is always hopping, so planning a short time in the line is never wise. The last time I arrived, in 2000, the line took an hour. But I've read of 15 minutes and 2 hours, so just roll with the flow.
You then pick up any checked luggage and go through customs, but unless you have something to declare, this is merely a matter of going through the "Green Channel - Nothing To Declare" line. Unless you are "randomly selected for additional screening" (which can happen but is rare), you can then go to a taxi.
BTW, a prebooked car (called a "minicab") will be much cheaper than a black cab, and if you pay cash, cheaper still. Just Airports is one minicab company often recommended, but there are others.
From London to Zurich, you will first go through passport control leaving London (in this case, it's emigration rather than immigration). Upon landing in Zurich, you go through passport control/immigration (again, same thing here), luggage pickup, customs, and exit. Again, trying to plan the timing will only cause undue tension. Zurich's airport isn't nearly as busy as Heathrow, and gets lots of flights from Schengen countries (these arriving passengers don't have to go through passport control), and so in theory should have a faster time in line. But again, how fast or slow the line is depends on how many agents on duty are processing how many simultaneous arrivals. Trains in Switzerland aren't usually reserved, so just get on the next one.
The US, by the way, is one of the few countries (perhaps the only one?) that does not have passport control on departure. So, flights to London can leave from the international or the domestic sections of O'Hare airport - check carefully where your flight is departing from so you don't waste time going to the wrong place. Unless you're flying back through Dublin or Shannon, however, your flight from Europe will land in Terminal 5, where the immigration and customs facilities are.
