That wouldn't be on account of the tubes; tube filaments do not take minutes to heat up the cathodes.

Warm boots can save time by not repeating all the boostrapping steps. In a warm boot, you typically don't execute any power-on self-checks, for starters.

Maybe I'm misremembering (I was very young) or maybe I was with people who were too superstitious, but these tubes themselves were actually from the 1950s, and they did indeed take a significant amount of time before they decided the hardware was "warm enough" that one could let the software start doing its thing.