BA are good at rebooking especially from North America. If you need to travel today I would examine the alternatives, find some combination of flights you'd be happy with (without regard for the price) and then telephone BA with your preferences in hand. They should accommodate you for free, but you will waive your right to compensation if you agree to a rebooking under your own preference.
The UK line is more reliable than the USA line. If your mobile telephone provider charges you extra to make international calls, consider using Skype-call instead.
BA will cover you onto American/US Airways/another BA flight/Iberia/Finnair without difficulty. They won't cover you outside that group without a lot of pushing (and to be honest it won't happen for a 24 hour delay).
In terms of compensation, yes it will be due (unless you negotiate your way out of the situation). They are also liable to cover your hotels/meal/local transportation. Personally, I find that BA is quite reasonable; I usually just submit a photocopy of my expenses and it gets paid back to me within a few weeks. We are talking Holiday Inn territory though, not Four Seasons. If you don't want to take the risk and have some time to kill, the check in desk should give you a pre-paid card to make such arrangements.