Theoretically not much. If one set of diodes is forward biased, the other is reverse biased with or without the bridge. There is a little junction capacitance added, but it's on the order of 4pF for 1n4148 and 15pF for 1N4004.
Paul Cochran, creator of the original Timmy pedal claims there is...
Anyone built this pedal? I know it's supposed to be blackface AIAB but the tone stack seems to be a scaled version of a tweed stack (Twin 5F8-A or Bassman 5F6-A) rather than any Blackface based on R, C and Potentiometer values, and mid connection.