Thanks for the tips everyone. And especially Oliver for this:
The 'route' you've linked to just goes down the A23. Fine if you want to do a spot of time-trialling, but not what I'd choose for a leisurely ride.
Exactly what I didn't want.
Ah, and I should have added that it leaves the A23 at Horley, but there are more pleasant ways of getting out of London.
@Oliver Schick started
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Thanks for the tips everyone. And especially Oliver for this:
Exactly what I didn't want.