In England, exchange means 'contracts exchanged' and both sides committed, or face heavy financial penalties. Before exchange in England, either side can pull out. Completion is the day that the buyer pays in full for the property (or the buyer's mortgage does), the buyer gets the keys and the seller has to vacate.
They are different dates.
In Scotland, once the offer is formally accepted both sides are bound. I don't think the Scots use the term 'completion' - they use 'settlement' and that is the day you move in.
If your Scottish offer is formally accepted, you are moving - so give your notices. Sounds like you need to check with your solicitor.