We use Merrill Lynch. The second year we were cruising I found myself given the runaround by the bank we had then, and we were almost without any funds over a long weekend in Venezuela. I was so angry with the hassles that we immediately started looking for a new way to handle our money. In 1987 ATMs were relatively new, and outside the US things were a lot more complicated. However, the switch to Merrill Lynch was a great idea then, and is still a great idea now.
We have a VISA card through Merrill Lynch, and once a month ML automatically pays the credit card balance off. We do not receive a paper bill, we never have late fees or interest, at the end of the year we receive a printout of every transaction, credit card, check, or wire transfer, that has occurred during the year. Our financial adviser is extremely helpful but not pushy, and will monitor our money if we ask her to. For example, we tell her we are traveling in the Caribbean for 6 months, and she will question any charge on the account that is not from where we are in the Caribbean (except for those payments that are recurring and usual, such as our phone bill, etc.).
We do not get our statements emailed - I understand that is a risky proposition. Instead, once a month I go to their secure website and download our statement onto my own laptop. They have a bill payment service (for those who have landside mortgages, etc.)
I understand that Merrill Lynch has offices in the UK, NZ, and perhaps other places as well, though I cannot speak for the services those offices offer. You do have to spend some time interviewing them in order to be sure it will suit your needs. There is a Merrill Lynch in Thailand, in Bangkok.
This has worked without a hitch for 18 or 19 years.