Update
So I turned up at Heathrow at 06:30 and asked to change to the 08:30 flight to YYZ. I first went to the check-in desk, and asked for a concierge, was told that there's no concierge available now, it's too early in the day, but one is in the lounge if I need to see them later. Ok, asked to do an SDC. Agent taps on the computer, and tells me: computer says no, fare not permitted to do same day changes.
Fine, I'm about to leave and head home, but stop by the ticketing desk. Ask the lady there and she taps on her computer, and says "no problem, it'll be £60". After it was all done, I asked her how come she managed to do it and the check-in agent didn't. She basically said that the check-in agents are lazy and don't know how to work the system, they're "petrified" of making changes. As the flight had only J buckets available, the ticketing agent said she just exchanged my Z fare for a J fare (don't know how that works) and then did an SDC.
In the future, I'll look for the concierge, if they're not available then I'll proceed directly to the ticketing agent. I recall when I've had success before it's been because the concierge was around and they would walk over to ticketing to make the change. I think the only benefit of preferring to deal with the concierge is that may be able to waive fees on the SDC.
Anyway, a happy outcome in the end. I now look forward to becoming SE100K at the end of the year so that I can email or call the concierge (but would anyone be around that early? Might have to still head to the airport and deal with ticketing if there's no concierge available until later in the morning).