I made a test in a lab environment, using the following devices:
I started an extended ping from R5 (ping ip SW1 si 1500 tim 1 re 5000), then I changed STP priority on SW1 to 4096. This is the debug log on SW2:
47w0d: STP: VLAN0001 heard root 4097-8cb6.4ff2.d000 on Gi3/1
47w0d: supersedes 8193-8cb6.4ff2.d000
No recalculation, no Topology Change Notice, no packet loss during the ping. The answer to the question is no, it doesn't have any effects on traffic. (Or at least no effect on lab traffic. )