Any change to the first segment should trigger a re-pricing and re-issue, not necessarily something you'd want to do without good reason. But afaik they can't tell you 'no' - I believe although ex-NA has to be ticketed 7 days in advance, the ticket remains fully refundable (is there a fee?) so you can always refund it and buy another.

You might find a BA employee who doesn't know the rules and lets you change the date, but in my experience with BA, their people generally seem business-like and deal with OWEs correctly. Changing the first flight changes the effective date of the ticket, a fairly significant detail, among other things.