Ticket code BPT is an Advance ticket. With Advance tickets, on paper the rules say you must travel from the booked origin station to the booked destination station, and that break of journey (which includes starting or finishing your journey at an intermediate station) isn't permitted.
However, in practice, tickets for Edinburgh Waverley and Haymarket have long been treated as interchangeable (equivalent to an unofficial "Edinburgh Stations" group) and so I wouldn't expect any difficulties starting your journey at Waverley.
In fact, in many situations you'd have to pass through two sets of barriers at Waverley to change trains when coming from Haymarket, so it wouldn't even be evident that you're "starting late" if you pass through a gateline at Waverley to take your booked LNER service.
After a
widely publicised story back in 2010 about a professor being penalised for getting off one stop early at Darlington, on an Advance from London to Durham, internal industry guidelines were written which still stand to this day and advise that no penalty should be imposed for break of journey on an Advance unless there is clear evidence of intent to avoid a higher intermediate fare.
Although of course we couldn't advise knowingly breaking the terms of your Advance ticket, it seems pretty unlikely that -
even if any member of staff somehow realised you were starting your journey at Waverley and took exception to this - you would be penalised in view of the above factors.
The worst penalty that could lawfully be imposed would be to have to pay the "excess" (difference) between the fare paid and the cheapest valid walk-up fare that permits break of journey. Seeing as Haymarket isn't included in LNER's
rip-off "Simpler Fares trial", the cheapest valid walk-up fare permitting break of journey would be the £91.20 Super Off-Peak Single for all London-bound trains after the 06:26 from Waverley on Mon-Thu - and for all trains on Fri-Sun. So in the worst case scenario you'd be liable for an excess of £54.30 - not ideal, but not necessarily the end of the world.
I think the only situation where I would advise caution would be if you were booked on one of the handful of direct Haymarket to Kings Cross services, as here it would be slightly more obvious that you were starting your journey at the 'wrong' Edinburgh station.