According to this resource http://lotrproject.com in The Fellowship of the Ring, Frodo leaves Bag End on September 22 and arrives (almost dead) at Rivendell on October 20 or 21. That same site has a neat distance time graph for both The Hobbit and Frodo’s journey in Lord of the Rings. http://lotrproject.com/timedistance/
Looking at both the site above and other maps, the distance to Rivendell to Bag End is about 420 miles as one of Saruman’s crows flies or 460 miles with assorted diversions. So the party of Hobbits went about 15-16 miles per day in that first part of the journey. Given the circumstance, the various diversions, and avoiding the main road, it is a decent pace.
The next stage of the journey has the Fellowship walk from Rivendell to the foot of the Misty Mountains. They leave December 25 and arrive at the base of Caradhras on January 10 having travelled 260 miles. That gives an average daily speed of 15 miles a day again. It looks like Tolkien used that as his rule of thumb for a kind of narrative speed with Hobbits.
Would they have got to Rivendell quicker with bicycles? Geographically, yes but narratively no. If it was a simple chase between Hobbits on bikes and Ringwraiths on horses then maybe with a sufficient head start, a cyclist could out-endure a horse rider (even an undead one given that the horses were spooky but otherwise still just horses). As it was, the Hobbits had riders behind them and ahead of them. It would be implausible that Hobbits could cycle faster at a sprint than a horse at a gallop if the Nazgul had them in their sights.
If, for some bizarre reason, we really wanted to speed up Lord of the Rings with better transport options then ships would be better. The Grey Havens are much closer to Hobbiton than Rivendell. It’s a bit further than Bree but not by a lot. Then it is a fast Elvin ship round to Gondor! Of course then the Hobbits and the ship-bound Fellowship get to have an encounter with pirates near the estuary of the Anduin. Presumably Gandalf would have been eaten by a sea-monster as they rounded Andrast (don’t worry, he’ll recover) and the whole of The Two Towers ends up back to front in this scenario. However, this version has pirates in it and given that Tolkien went to all the trouble of adding pirates as a plot point to the book then it would make sense to actually meet some pirates.