Here in the desert, we dig in organic matter, sulfur, and gypsum to keep our "adobe" clay loose. In the UK, you might want to replace the sulfur and gypsum with coarse limestone, especially if the soil is acid. "Sharp" sand can help, also, but it needs to be partnered with organic matter, such as compost, or the soil turns into puddled clay.
If you are talking about breaking up the large clods from the first digging, let them dry out a little, and then turn them again. Somewhere between "doughy" and "rock hard", there is a moisture level called "friable", where the clods break up into crumbs when struck. Using a spading fork, rather than a spade is also helpful, since the fork tends to break up the clods, while the spade just polishes them.
Good luck, and remember to pace yourself!

Hello,
You probably don’t want to hear this now but you’re really lucky. When you get your clay soil in hand it will be moisture retentive, fertile and support a really wide range of lovely plants. I have had heavy clay soil in my past two gardens. The way I got it in better shape for planting was to start small. Don't go mad with digging a large area. Start with a small part of the border. Dig down as deep as you can, about two spades deep should be enough. Fork over the bottom of your hole as much as you can to help drainage. You might find that you have a solid pan of clay around this depth. Replace the soil and incorporate as much organic matter as possible. Homemade compost and plenty of manure is ideal. (I tried grit and sand but found that my clay soil ate it up and it disappeared and it’s costly). The next bit is easy. Don’t ever dig it again. You’ll just bring more clay to the surface. I mulch mine every year with homemade compost either in the spring or the autumn. Best of Luck.

my garden was as hard as a rock when I came here!!! Agree with Julia. Lots of organic matter, lots of manure and then lots of compost. Mine is tamed now lol

Sand has worked for me, but I have been using it at a rate of 1 part sand to 3-4 parts "adobe".