As Steve and Tom said, you may find some merchants in England that don't want to take the Scottish notes. As I understand it, it's less hassle the closer to the borders you are and the further you get from Scotland the greater the chances that they will be rejected.
When I went to Scotland last year I had a few Bank of England notes with me at start, those spent fine in Scotland - no issues whatsoever. I then started to get Scottish notes from the ATM and as change and once I got to London I still had a few of these. I didn't encounter any issue spending the few I had in London (some in a cab and at breakfast one day), but your mileage may vary.
Your best bet is to probably try to use up most of the Scottish bank notes before you head south just to avoid any hassles.
If you want more info on the ins and outs of the situation you may want to Google it. It's kind of interesting in that the Scottish notes are not even technically legal tender in Scotland (only the coins, which are same in England and Scotland are true legal tender). But the law allows leeway and gets around the issue.
You may even, for fun, decide to keep some of the Scottish notes as a souvenir as the 3 banks (RBOS, Bank of Scotland and Clydesdale) all have their own notes, with different art on them.