I just arrived to Zürich on a work visa (US citizen) and will be applying to at the kreisburo receive our work permits + residence cards next week after I receive my permanent rental application.
It is my understanding that it will take 2-4 weeks to receive my physical residence card. During this time is it possible to travel to the UK or countries in the Schengen area? I have a multi-entry D visa in my passport that is valid until the end of the year.
Additionally, does it typically take the full 2-4 weeks to receive the resident card, or does this happen sooner sometimes?
In general, paperwork moves fast in Switzerland and it is very likely that your card will arrive faster than the worst case estimates.
The proper thing to do, if you want to travel in the meantime, is to ask the local authorities to put a temporary, multi-entry Visa in your passport(s), based on your pending permit application. There will be a fee. In theory, this is necessary, because the D visa only lets you enter once, and you aren't supposed to enter as a tourist (sans-visa or permit) if you intend to stay. If you need to travel right away, you can ask when you turn in your paperwork for the permit. But the timeline to get it really might just be a week or so, so if your feet aren't too itchy you might wait and see.
Also, in practice this might be overkill for a short overland trip to a neighboring country. The odds of a passport check are low, and if one happens the odds of real consequences are also low e.g. a simple scolding is the typical consequence if you fail to show your permit with your passport when going through passport control at the airport.
You can get a paper saying your permit is being processed and since you’re non-EU the approval for the permit will already be on record with passport control. However, I would not advise travelling until you have at least done the biometric part of the permit. You will be notified when you need to do that part of the process.