Question: What are some real world situations which can be modeled using uniform random variables?
Answer: Consider Births throughout the year. It seems likely that all days of the year would be equally likely. Thus the uniform model for birth would say that we can model the birth of a child in a given year as a random variable which take all of the possible values 1,2,..., 365, with equal probability.
EXCEPTION: Some births are scheduled -- for example by Cesarean Section. If the doctors schedule these by convenience, then these would be more likely during office hours on Mondays to Friday, and less likely on holidays. This might cause some change from the uniform distribution.
Answer 2: Consider glancing at your digital watch, and noting the number of seconds it shows. It seems likely that this would be uniformly distrbuted between 0,1,...,59.
ON THE OTHER HAND: If we look at the HOUR at random, this may not be random, because we would not be looking at the watch during our sleep time. Some hours would become less likely, while others would become more likely.