When you activate the Phylarch, you will only receive bonuses from those Skills assigned to the active Skill slots.
In the Phylarch tab, you have up to 6 unlockable slots. You may select 1 Phylarch Skill in each. Once you activate your Phylarch, the selected Skills will start providing you with the appropriate bonuses.
Percentage-based Phylarch Skills, which accelerate your Agreement 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 Phylarch with a Unit training speed bonus applied, the remaining time of all your training queues is recalculated, taking your new training speed into account.
For example, let's say it takes 2 hours to train 100 Swordsmen. Then your normal training speed is 50 Swordsmen per hour. With the 20% Phylarch bonus active, your training speed will increase to 60 Swordsmen per hour, so with the Phylarch bonus activated, it will take only 1 hour and 40 minutes to complete the same 100 Swordsmen training.
On the other hand, if the Phylarch 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 Phylarch bonus is active, your Building will be upgraded at a higher speed, letting you complete a certain percentage of the upgrade (let's say 70%) faster than you normally would. However, once the bonus expires, the time needed to complete the remaining 30% of the upgrade will be recalculated once again.