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If they do not bloom, it will mean that they bloom on old wood, and likely need several years of unpruned or unharmed growth to flower.

https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/hydrangea-not-blooming.htm

It would be best to wait till fall and when it isn't blooming. Since it needed to be moved within a few days, just make sure to get most of the rootball and have the new location dug and waiting. Make sure it is well watered till it gets reestablished.

The roots are composed of underground rhizomes that elongate and spread out as the plant grows. Typically the root zone is as far out as the canopy.

https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/transplanting-hydrangea-bushes.htm

This reminds me of the variety called the Lacecap!

Here is an article that will help:

https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/lacecap-hydrangea-care.htm

For both plants, you only need to remove dead wood with some regularity. In the south, pruning of dead wood happens after the end of May i case the stems are alive and leafing out late. Further north, you can wait until June or even July. If you need to prune live wood in hydrangeas that bloom on old wood, do so after the plant stops opening flower buds. For hydrangeas that bloom on new wood, avoid pruning somewhere from mid spring to as late as late summer/fall as that is a period when those hydrangeas develop flower buds and open flowers. Rhododendrons should also be pruned after the flowers begin to fade. Removal of the spent flowers is what is called deadheading and it can be done at any time. Deadheding hydrangeas does not encourage reblooming as you get a maximum of one bloom per stem per year with any hydrangea. It typically consists of short cuts of the stems (say, the first set of leaves); you can also cut at the peduncle, the string that attaches blooms to the stem. If the cut is short, you do not interfere much with any flower bud development that could be going on.

These vines can be pretty particular, unfortunately. There are a few things that you can do to help it along and get it to bloom better, though. This article will offer suggestions:

https://www.gardeningknowhow.com/ornamental/vines/climbing-hydrangea/climbing-hydrangea-wont-bloom.htm

Hydrangeas can fail to bloom for many reasons. Not knowing what type of hydrangea it is and it’s cultivar name, consider and research these possibilities: growing in dense shade; applying too much fertilizer; growing out of zone; pruning at the wrong time; winter or frost damage/injury to stems; pests that eat the flower buds; etc.