Hi,
I haven't been looking with any depth recently, but I don't think any bank based either in the UK or USA, will allow/have the option to keep a US Dollar account with a GBP one (or vice versa) when using one of their basic bank accounts. This is down to the fact that no US based bank has UK branches here, and vice versa. Although 'HSBC' has a presence in both (and other countries) the representation in each country is it's own entity. You could argue that they would or could share account information etc, but having worked within the IT infrastructure side of things at some of the biggest financial institutions in the world, it's not as 'simple' as that. Leading on from this, there's also actually extremely little demand for what you'd like, there's really not that many numbers of people actually moving around the world and relocating etc when you take into account population sizes.
I think that if you go up to higher level bank accounts, you might be able to get some sort of connections, but you'll have to ask each banking institution if they have any such functions/capabilities etc.
Lastly, you've sort of answered your own question, it's now so easy to login online to control accounts from anywhere in the world, so there's even less 'demand/reason' for a bank to operate in another country or offer the above aspects in basic bank accounts - do check however with higher level accounts. I'm not American, yet last year I walked into BoA, Chase & Wells Fargo and opened up basic accounts within 25 minutes and I can control them from here. The only aspect is I need to login to each bank's website.
Cheers, DtM! West London & Slough UK!