There have been a couple of apocalypse-in-progress novels, usually where an EMP (Electormagnetic Pulse) or solar flare has (realistically?) knocked out all the automobiles. The hero decides (so far they have all been men) decides he needs to get somewhere a long ways away, and uses a bicycles.
Just how far can you travel by bicycle?
According to Wikitravel, you can feasibly make 250 km (155 miles) per day, but that 120 km (74 miles) per day is much more realistic. They note that you can generally you can make about 25 km (15/5 miles) /hour, but with panniers (saddlebags) this is likely to be reduced to 15km (9 miles) /hour.
They note that going up steep hill can reduce your speed to 1/10th of normal, and that you are often better off going longer less steep routes than shorter routes that are very steep. Walking normally is about 5 km (3.1 miles) per hour. Obviously in difficult terrain, where you have jumped off your bike and pulling it alongside you, that is going to be a greatly reduced number.
Bicycles were extensively used in both World Wars - occasionally with scouting units, but mostly for couriers. There are still people today who are interested in their military application. Note that military units, with their general youth and training can make much better progress than you typical civilian.
In an account of the 25th Infantry Regiment;s Bicycle Corp:
1896 African American Buffalo Soldiers test bikes for Army on 1,900 mile expedition
Eric Shalit, Tubulocity, 17 January 2011
The following summer, 1897, came the Bicycle Corps’ most remarkable adventure, a 1,900-mile trip from Missoula to St. Louis, Mo. In 34 days of riding, 20 soldiers averaged 56 miles per day. Their average speed registered 6.3 mph. Newspapers carried daily updates on their journey, and the Army & Navy Journal quoted Lt. Moss at the conclusion:
Note that they were carrying 120 pounds of supplies, and some of the terrain (it included parts of Yellowstone Park) were very rough. Of course, I also expect that the troops involved also felt they had a bit of a point to prove, so I think we can say that their performance while certainly duplicable today, would be exceptional.
In net, I have to say that the bicycles looks like a very attractive alternative. While it is true that you are more vulnerable on bike, some of that vulnerability is reduced by the fact that you're overall exposure time is reduced.