We have lived on a 45 footer for 2 1/2 yrs including last winter in the Eastern Caribbean and we are starting a circumnavigation in the fall. We tend to live quite simply on the boat - for example, no nights in marinas in last 6 months, eat out about once every 2 weeks on average and eat whatever is available locally at a decent price - sometimes this is quite like North American food, sometimes not, either is fine. Our boat is quality-built but almost 30 years old. We spent close to $60k getting it ready for extended cruising (don't assume that because the boat you are buying is newer that there will not be considerable expenditures to be made, but that is a different topic.
So much for the context. Our spending in Caribbean averaged about $1200 a month not including insurance. One month it was closer to $4000 because we decided to upgrade the windlass and get a new laptop and prices in St Martin were excellent. We had no significant breakdowns on the boat with something over 6000 miles travelled so boat expenses each month were modest. We did have contingency funds available but did not to use any. People mention $100 /ft/month, in our case that would be $4500 per month. I don't think that anyone we met in the Caribbean were spending nearly that.
Cruising in the Med and other parts of western Europe is a different matter entirely. Many of the Europeans we met in the Caribbean said they were there because sailing was so expensive in their home countries. For example, apparently there are no (as in none) places to anchor on the Med coast of Spain and dockage for a 40 footer for one night is generally around 70 euros. Also other things are very expensive as well compared to the compared.
You also mention going back and forth across the Atlantic. This is not a casual undertaking and will be hard on the boat even with crossings in the best seasons and best places. Most people sail to Europe and stay there for some time ( 2 to many years) before coming back to the Americas. Our original plan was to sail to Europe this summer and stay there for a couple of years but it seemed to be out of our budget range.
BTW, if you do go, it is possible to winter in the downtowns of both London and Paris for not too much money.
I would encourage you to pursue your goal but I think you need to do a great deal more research.
Finished the circumnavigation in early February in Grenada. Have to work on a book project for the next several months so the boat will be waiting for next year.