[Paul]

BT systems control your line speed as Sky Connect is a just rebadge of BT Connect. You are on BT exchange equipment running BT tools, Sky have no actual hands on control of your line setup but they are causing the throttling in a different way....
Sky only bought enough bandwidth for a set amount of users on Sky Connect so the service is badly oversubscribed which is why the speed dies at set times of the day.
Like I said it's coincidence that your speed dropped when it did as Sky cannot tell BT to drop your speed, (they can ask for your line to be re-profiled but that's not the same and takes a while to actually happen) you hit a heavy usage period and the speed dropped due to the subscription issue.
The BT guy is correct in saying BT do not throttle but didn't explain why or how Sky is causing the throttle effect to occur and it's just down to over-subscription and not enough bandwidth being paid for. They are buying more bandwidth soon but I would still not agree to take Sky connect no matter what sky offered with it.
Speed negotiation on Connect (Sky or BT) is automatic but with BT Connect you won't get the contention and over subscription that you get with Sky Connect and so should see your speed achieve the maximum the line can take.
Do you know what your current stats are as often BT underestimate what speed a line can achieve?