I think there's some risk to heading to Scandianvia, Finalnd and Estonia in September, especially in the second half of the month, but it could turn out fine. You can find actual, historical, day-by-day weather statistics on the website timeanddate.com. I find that sort of detailed data a lot more useful than monthly averages. I want to know how bad it has been over the last five years, not that the average high temperature (meaning around 2 PM) was decent. (I may care about the temperature a lot more than you do.) This link will take you to Helsinki's weather in September 2022. Use the Search box near the top right to change cities. Use the pull-down menu just above the graph to change the month and year.
Keep in mind that monthly hours of sunshine are likely to decrease rapidly through the month of September. That's one of the statistics included in the climate-summary charts in the Wikipedia entries for many cities.
The northern destinations would be my last choice of the three options you present for the month of September.
A car would be helpful in getting to some places in Normandy, but I managed to see all of the following by train and bus: Honfleur, Rouen, Falaise (WWII museum), Bayeux, Caen, Cabourg and Deauville. There are small-group tours offered to D-Day invasion sites if you want to see them and don't want to rent a car.
Public transportation will get you to Biarritz, Bayonne (which I much preferred) and St-Jean-de-Luz (also more interesting to me than Biarritz, but there was a weather differential at play). To go to some of the small mountain villages in the French Basque Country, a car would be very useful.
There's very, very fast train service from Paris to Bordeaux. Bordeaux is seen by many people as primarily a jumping-off point for the Dodorgne and Lot rather than as a destination itself, though I enjoyed the historic section of Bordeaux. A car would be extremely helpful outside Bordeaux unless you're satisfied to stick to cities like Cahors and Perigueux and forego visiting the charming villages in the area.
I like the odds of decent September weather in southern England a lot better than in places like Finland. I've been to England several times in September and haven't had a problematic combination of cold weather and rain. I believe rain is likelier the farther west you go.
If you want to do more than spend an hour or two in each of a bunch of cute little villages (doable via small-group tour at modest cost, with some tours originating in Bath), a car would be pretty essential in the Cotswolds. Moreton-in-Marsh has train service (good links to Oxford), and there are some buses fanning out from there, but bus service seems so infrequent it would probably be difficult to visit more than one village a day. I've read on this forum that taxis are not widely available.