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Make sure that the soil drains welll in the new location and has no nutrient deficiencies. If all the new spring growth originated strictly from the crown/base, the new location is exposing last year’s stems to environmental conditions that are not favorable and which kill or dry them during winter. Winter protection May be necessary to regain reliable blooming. Or, you could try a rebloomer hydrangea from the Let’d Dance Series (choose cultivars that do not grow taller than 3 feet though), Smooth Hydrangeas (aka, hydrangea arborescens; morning sun only until 12pm) or Pee Gee Hydrangeas (aka, hydrangea paniculata; can get full sun).

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