Western Europe is definitely a possibility, because people RV there, so there is an infrastructure to support that activity. The facilities sometimes resemble our RV parks (although usually more crowded) but more often a "campground" will be more like boondocking, but usually boondocking for a fee.
The difficulties have to do with how long you can stay as a visitor, and the fact that you are moving around some rather expensive dutiable property. There are ways to work with or work around the customs and immigration restrictions that tend to keep our travel short (often 90 days or less) and make movement of expensive assets problematic.
Some places may let you stay as long as six months as a tourist, and it is not as difficult as formerly for U.S. citizens to move around among countries within the 26 Schengen countries (a subset of the E.U.), but you still won't have the freedom of movement that is the privilege of an E.U. citizen.
Customs fees can be more of a problem, because posting a bond or paying the fees in one country may not cover movement into the next. There are assumptions about what goods are being brought into a country temporarily for the visitor's personal use, what goods are being imported. Motor vehicles from outside the customs union, including RVs, tend not to fall into the first category.
These things can be dealt with, it is much like the problem yacht and private aircraft owners have when moving from port to port, country to country. You can travel all over the world in a RV if you have enough money, patience, and persistence to deal with officialdom.
There are also technical issues. Most RVs sold in the U.S. will have the wrong electrical and LPG systems for the RV infrastructure in Europe (or the U.K.) and things can get more strange as you move beyond the harmonization cloud that is the E.U. There will likely be safety equipment issues, but those modifications are mostly lighting, and relatively simple to have done.
The next problem can be size. Our RVs tend to be large, some as large as motorcoaches, but motorcoaches do get around in Europe, they just can't go all the places smaller vehicles might go. The greater problem with our large RVs will be dealing with the licensing and road tax issues. In most of Europe, anything longer than seven meters, anything heavier than 3500 KG, is a heavy goods vehicle (sometimes presumed commercial), with operator licensing restrictions and special operational requirements. What these are, and what are the thresholds, still varies country to country, so a large RV that is OK in Germany might not operate under the same rules in Italy or the Low Countries.
The best introduction to RVing in Europe might be a short trip, 2-3 weeks, renting a European RV. You will get a feel for the RV lifestyle there, what kinds of places there are to stay, where you can and cannot go (even a European-size motorhome is restricted from many small destination towns), and what it costs. The rental companies will usually take care of the "official" things for the countries where that rental is contracted to go (they do even more of this for escorted tours) but you can get a feel for what it is like moving country to country.
I do see U.S. built Class A motorhomes running around in Western Europe, RVers there like them, import and convert them, and deal with the size issues. Some of the issues are a matter of cost, and if you are wealthy enough to pay the fees... The operator licensing issues are easier for European residents to deal with, they can get the training, pass the medical exams, get the heavy vehicle licenses. This part is not so easy for visitors, who are usually driving on non-commercial licenses issued in their own countries, accepted as a matter of treaty or courtesy.
