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That's pretty much my commute. Lots of traffic and traffic lights though.
Would be quieter to carry on Fleet Street until Shoe Lane and then head through smithsfield and golden lane estate.
Although having said that, the last time I tried it there were building works blocking part of shoe lane. You could go up new fetter lane instead although you'll hit more traffic and a big set of lights..
I'm guessing you go along below the river and then over London Bridge and up Bishopsgate? Quite traffic lighty that way.
My preference is to go across Waterloo bridge, right over Aldwych, down Fleet street for a little bit, up Chancery Lane >> Gray's Inn Rd >> then just down Clerkenwell Road as I find it's quite quick usually.