Assuming US passport and purpose of travel is only tourist…

Your passport will be stamped at first entry into each border zone. UK Customs and Border will greet you with a brief interview about the purpose and length of your stay, lodging, and onward travel arrangements. If you have a satisfactory interview, then you can be granted tourist entry up to 180 days in a 360 day period.

When you first enter the Schengen border zone you will likely just have to show your passport, it will be stamped, and you are typically allowed up to 90 tourist days in a 180 day period.

If you’re schlepping a huge amount of luggage, are unclear about the reason and length of stay, you will likely be questioned further and asked for proof of sufficient funds, insurance, lodging arrangements, and proof of onward travel within 90 days. (180 days UK)

Either way, it will be stamped at each first entry. UK and Schengen are different.

Airlines are not even supposed to check you in if you don’t have proof of onward travel within the allowed tourist entry days for your destination.

Sorry to rain on your parade, but it’s better to get this sorted in advance, rather than face fine, deportation at your own expense on arrival, and ban from re-entry.