Money/ATM/Credit Card in UK, France, Netherlands . Several Questions

I think the Capital One Visa card will not charge a foreign transaction fee every time it is used. Mine doesn't. I don't know if all Capital One cards have the same terms, at least as regards foreign transactions. The Chase card may charge a fee. We'd need more details to know for sure. (Does the Chase Sapphire card not charge a fee? Not sure.) What type of Chase card do you have? As for the ATM, the same question applies. Is it a BOA card that is exempt from fees. I have a BOA credit card that doesn't charge fees (a holdover from when it was a Schwab-branded card) and a BOA credit card that does. You can probably use the Visa for bigger purchases if there is no foreign transaction fee and if the charges are made in the currency of where you are (don't fall for the semi-scam of converting the price to dollars before you buy). It may not make much of a difference to use credit versus pulling cash from your checking account.

Why not open a Hi-Yield Money Market Account at Capital One? Then you get an ATM (not Debit) card that Cap One charges no fee for use, and the European bank machines won't charge for use either (do not know the UK rules). You can get all the cash you want at no fee, and the exchange rate for using the Cirrus network will cost about .8% over the published exchange rates. Many smaller places do not want to take credit cards, and even if they do, you are doing them a favor by letting them avoid their credit card fees, which they might even pass back to you with a discount. My Cap One Mastercard does not charge the foreign fee, and I appear to get the published exchange rate Of course, just yesterday I posted a similar question about ATM fees in Canada and was met with a lecture as to why I should just give the banks their fees.

BOA has partner banks in several countries (BNP Paribas in France), and if you use their ATMs, BOA charges no fees. Check the BOA web site for a list of such banks.
Nor does Capital One charge a transaction for their credit cards, and most places do take credit cards, even American swipe cards, although I understand Amsterdam may be an exception, as are unattended gas and train station machines.

Since your only ATM card is is with BoA, I would start there. Ask them what fees they charge with international use. Just be careful how fees are expressed, All cards will be hit with network fees that amount to ~1%. Depending on your Bank these may itemized and show as a Currency Conversion Fee and a Foriegn Transaction Fee, or they may just bundle it into other fees, or just show the transaction amount. On top of all of that, BoA may charge an additional fee that is all their doing, and since they advertise "no Fees" at some European Banks, I suspect they have a charge. The exchage rate you get with the ATM, will actually be very close to the advertised Interbank Rate, the rate you see published everywhere. The "public conversion" rates you refer to are probably the rate for exchanging currency at your bank here in the US, that will not be what you pay. For your credit cards, the exchange rate will be the same as the ATM, but fees will be higher in most cases. Usually I figure that the ATM gets me (in the case of my cards) exchange for about 1% or less, if I use a credit card, then it costs me more like 4%.