We moved to a new build just after we were married and there is no quick fix. We spent the summer removing rubble, marking out borders and digging in as much organic matter as possible. Bags of compost, mushroom compost, and anything to break up the clay. We laid a lawn and put in a patio. If my memory serves me well it took about seven years until it was a proper garden. It was well worth it.

Sounds as though drainage is the only answer if its as bad as that. No quick fixes, sorry.

Anything you put on to absorb the present flood will only help until the next rain, when it will overflow. Also, a layer of organic material will start to smell if it remains saturated for long. Long term, I would get a soil analysis to find out the pH. If acid, apply the recommended amount of dolomitic limestone. If alkaline, apply gypsum, instead. Also apply about 5-8 cm of compost, and 3-5 cm of grit, and till it all in about 30 cm deep. The tilling is best done when the soil is dry enough to crumble instead of squish when squeezed.
Concerning the drainage system, where would the water go? If into your neighbor's yard, does he/she know? If one corner or edge of your property will remain swampy, look into riparian plants that can absorb and evaporate the excess, such as River Red Birch, or Blue Alaskan Willow, and/or a lily pond with a puddled clay bottom.

It's the housing company's responsibility to take care of this problem in total. Keep good records including photos of this and make no attempt at amending this problem or you will take the blame for it or the blame for their failure to correct it. Send them a registered letter return receipt requested acknowledging their intent to resolve this problem( I hope this was done in a letter to you on company stationary) and request that it will be done in a most timely manner. Again, do not lift a finger to solve or ameliorate this situation on your own for it will adversely affect you if it comes to a legal fight to get things done correctly and to your lasting satisfaction.

Yes, I should have emphasized that all of my recommendations should come after the housing company has done their work, just to ensure an even lawn, healthy flowers, robust trees and shrubs, and good relations with your neighbors.

While this is getting corrected you might look into "lasagna gardening" which is one way to plant when there is heavy clay soil to contend with. I used this method with mine and after a few years I achieved great results.

I'll have to review the instructions. The one time that I tried "lasagna gardening", all I wound up with is more drainage problems. That was some 25 years ago, though, and I might have done it wrong.

Well that could be Tug. You see my property gently slopes down into a protected woodland within which there is an artesian spring.

The builders had to put in a French drain along the upper border of our garden as all of the houses had a stream running along the back when we had a huge rain storm - that worked but we still had to improve the soil.