Some neuronal types (from Cajal)

Your question, "why reflex times change with age", goes to the core of how the nervous system is established and how it processes information.

The human brain develops through a number of stages.  Neurons are cells, but they are not themselves capable of dividing. 

Neurons are said to be "born" at the cell division that produces them.  Neurons are born at various points during fetal (before birth) and neonatal (after birth) development.

There are many different types of neurons – they differ in where they occur within the nervous system, in their shapes, what chemicals they respond to, and what chemicals they use to influence other cells. 

At their birth, a neuron generally has a fairly simple shape, but as it matures, these shapes can become quite complex. 

A typical neuron consists of three distinct cellular regions, which are called its dendrites, its soma (or cell body), and its axon. 

While synapses can form anywhere on the neuron, they typically form on the neuron's dendrites.  While the neuron's axon makes synapses on other neurons. 

The process by which the human nervous system is " connected" is complex and takes many years.  It begins with many more neurons being born that well eventually survive and many more synapses being made than will remain in the mature system.

There is a competitive process of pruning incorrectly connected neurons and synapses.  As this occurs, neurons become appropriately connected and "extra", inappropriately connceted neurons die. 

This process is driven by activity and leads to a well coordinated neural system.  

This is one of the reasons why practice makes perfect!   The more you practice a task, the more you remodel and reinforce the neuronal circuits involved, and the better you get at it.

This is also the reason that we can learn new skills as adults, the process of neural network remodeling keeps going on until we die.

The final process involved in the maturation of the nervous system is the process of myelination. 

Myelin is an insulting cellular sheath that surrounds many axons.  It acts very much like the insulation around an electric wire.  The process of myelination proceeds for decades after birth in humans.

The more myelin, the faster a nerve can carry an electrical impulse.  The coordination of the nervous system depends on the speed that signals travel, and it takes time to "tune" the system.

Most activities depend upon coordination between senses (sight, hearing, etc) and movement.

Disruption of myelin, as occurs in the disease multiple sclerosis, leads to discoordination in part because signaling speeds within the nervous system change in unexpected ways. 

So, the answer to your question is that, during childhood the nervous system is not yet optimally connected -- such a well-connected nervous system takes time and practice. 

As we age (or stop practicing) cell damage and disuse again leads to the loss of neural coordination.

While it is common to think of reflexes in terms of physical activities (catching a ball), the same is true with original and critical thinking – as with the physical it is important to keep your mental reflexes sharp by practice (and that is why, at least some older people tend to be wiser than the young!)