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These are usually pruned back in the fall, though it should be ok to prune back some in spring if needed. This article has information on pruning hydrangea plants: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/prune-hydrangea-bushes-hydrangea-pruning-instructions.htm

Here is an article that you may find helpful: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/hydrangea-not-blooming.htm

There are several things that can affect the blooming. This article can help: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/hydrangea-not-blooming.htm

I would give it a phosphorus heavy fertilizer instead to promote blooming.

Various fungal spots can develop on hydrangea leaves and are normally triggered by too much or too little water. Spots also develop when hot sun strikes wet foliage. Water only when the soil is dry and use drip irrigation, if possible. You want the water to soak into the soil and not get on the foliage, which can lead to these fungal problems. You can also retain soil moisture by using mulch. For the fungus, treat the shrub with a neem oil foliar spray.

The soil around the non-blooming hydrangea may be lacking in phosphorus. That can affect the ability of the plant to bloom. These articles will help you learn more:
https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/hydrangea-not-blooming.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/phosphorus-plant-growth.htm

Climbing hydrangea sometimes can take a few years to get blooming after being plants. I would give it some phosphorus though to see if you can get it jump started. Bone meal is a great source of phosphorus.