The timer values are sent in the STP BPDU packets from root towards the downstream switches. The downstream switches will use those values configured on the root.
When you are checking the 'show spanning-tree' output, there are 2 sections, one starts with Root ID, other starts with Bridge ID. If you have changed the hello timer on the root, downstream switches should indicate that value at Hello Time in the Root ID section and their own default in the Bridge ID section.
Root ID Priority 32921
Port 2 (GigabitEthernet1/0/2)
Hello Time 1 sec Max Age 20 sec Forward Delay 15 sec
Bridge ID Priority 32921 (priority 32768 sys-id-ext 153)
Hello Time 2 sec Max Age 20 sec Forward Delay 15 sec
Aging Time 300 sec
Note that Hello Time is different in the 2 sections. Also, they will not propagate this value to their running configuration.
Regarding the effects of changing the timers, I would recommend you to check the following guides:
Understanding and Tuning Spanning Tree Protocol Timers
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