First of all, please note that when you activate the Dragon, you will receive bonuses only from those Arts that you assigned to the active Art slots.
In the Dragon tab, you have up to 6 unlockable slots, each one of which you may select as 1 Dragon Art. Once you activate your Dragon, the selected Arts will start providing you with the appropriate bonuses.
Also, please note that percentage-based Dragon Arts, which accelerate your Lost Arts upgrades, building upgrades, and troop training process, actually increase the upgrade/training speed rather than cutting off a certain percentage of the time.
When you activate your Dragon with a unit training speed bonus applied, all your training queues remaining time will be recalculated with your new training speed taken into account.
For example, let's say it takes 2 hours to train 100 Pikemen, and your normal training speed is 50 Pikemen/hour. With the 20% Dragon bonus active, your training speed will increase to 60 Pikemen/hour, so with the Dragon bonus activated, it will take only 1 hour and 40 minutes to complete the same 100 Pikemen training.
On the other hand, if the Dragon bonus expires before the training queue is complete, the time needed to train the remaining units will be recalculated once again, with the decreased speed now taken into account.
The exact same principle applies to the building upgrade process. While the Dragon bonus is active, your building will be upgraded at a higher speed, letting you complete a certain % of the upgrade (let's say 70%) faster than you usually would, but once the bonus expires, the time needed to complete the remaining 30% of the upgrade will be recalculated once again.