I admit, twenty miles is not very near, and the novel does not say who owns the living at Woodston, either.

On the speed of carriages:
We know that Henry's curricle travels faster than the Generals chaise ("so nimbly were the light horses disposed to move, that, had not the general chosen to have his own carriage lead the way, they could have passed it with ease in half a minute"), and it is reasonable to suppose that it travels faster than John Thorpe's gig (as James Moreland scientifically demonstrated in chapter 11, that moved at 7¼ mph, tops).

The Curricle was a two horsepower machine, and infamous for accidents. They were fast, if one horse stumbled they both fell; they had a short wheelbase (or maybe that should be hoof-to-wheel base), were high set, and were very light - so they accelerated quickly, and would therefore spin out and tip over just as quickly- although there were also writers who spoke of the curricles superior safety and easy handling. The Prince Regent had one.
It was also infamous in fiction, as the carriage that unfortunate heroines are whisked away in, that dastardly villains are overturned in, or inconvenient heirs unexpectedly run over by; although, (as far as I can tell through Google searches)as a literary device, the curricle seems to be more popular in the 21st century than it was in the 19th.

References
Discovering Horse-drawn Vehicles By D. J. Smith(Osprey Publishing, 1994)

A treatise on carriages: comprehending coaches, chariots, phaetons, curricles, whiskeys, &c. : together with their proper harness ...
By William Felton (Published by the author; London, 1794)