Where do you live? And what is the tree's variety name?

Beyond that, it's best to remove fruits the first 2 to 3 years to allow the tree to grow a sturdy root system.

Tim, this year we had an unusually cold winter.

Fruit production for Lapins is off and on some years in the coastal Southern California area. By the fourth year, fruit production on Royal Lee/Royal Minnie Lee is pretty good. Still a bit too early to tell though because most people haven't had Royal Lee that long (since the variety has only become available relatively recently).

I think the cherry tree being partially shaded in the winter really makes a difference. I have a Rainier (first year, planted in a pot) and in early March it was loaded with blossoms. Rainier's not even considered a low-chill variety. And I have a Royal Lee/Royal Minnie Lee pair, full sun, second year in the ground, no blossoms (but leafing out well).

My other Minnie Lee (yes, I have too many cherries) in a pot in the shade was also packed with blossoms; it blossomed about a week ahead of the Rainier (but the blossom time did overlap). Even my Bing (yes, I know) had a few blossoms form on its tips.

Flowering is a very good sign, but that still does not absolutely mean the tree is going to have enough vigor to set fruit. (I'm talking about stone fruit trees in climates that may not have quite enough chill)

In some cases some fruit may form but it will end up dropping off the tree before it gets very big.

You know, I've been wondering. What if I grafted a branch of Royal Minnie Lee onto something like Bing? Would that be able to help bring the Bing out of dormancy? I've noticed that different parts of the tree seem to operate independently at first, but then when one part of the tree begins leafing out it seems to initiate the surrounding parts of the tree to quickly begin leafing out soon thereafter.

Are you willing to experiment on your Bing, being it is the only one? If so,

Try it. If you don't get the results you want you can probably cut it off and return to present habits. The worse case is a dead tree.