Chris,
I'm not sure the information provided by your Banker is entirely correct. I just returned from Europe a few days ago, and used a magnetic-strip MasterCard extensively for ATM withdrawals in France, Italy and Switzerland. The card worked without any problems. If you use a card which is affiliated with your financial institution (Cirrus, Plus, Maestro, etc.), the card should work.
One point to note is that you probably won't be able to use your normal card for "point of sale" (debit) transactions as you do at home - only ATM withdrawals.
I haven't been in the U.K. since late 2007, so can't comment on the current state of ATM machines there. I'd be very surprised if the ATM's there will ONLY accept chip cards, as there are still many tourists that haven't got a chip card yet.
I also have a new Visa card and this is equipped with both a "chip" as well as a magnetic strip. I didn't use that for ATM withdrawals though, as they would have been treated as a "cash advance" with horrendous interest rates! I did use it to pay a few Hotel bills and purchase rail tickets from a machine - no problems at all.
It would probably be a good idea to have at least one "chip card" if you have time before your trip, as these are becoming the most prevalent card in Europe. The days of the magnetic stip card are definitely limited!
One other point to note to note is that if you'll need rail tickets from SNCF machines in France, these will ONLY accept "chip cards". As of a few days ago, the ticket machines in Milan were still accepting magnetic strip cards.
Happy travels!