I've been doing a very similar project, and I've found the best way of getting rid of brambles is to cut off the top growth, as you have done, leaving a short "handle". Then you use this and a fork, crowbar, spade or whatever tools you have, plus brute force and ignorance and heave out as much of the root system of each plant as you can. It is not possible to get every fraction out, so don't stress about it too much. Now you wait a few weeks and go round again, looking for regrowth. All the bits of root you couldn't get out start producing leaves - and get less deep in the process. The bramble plant seems to "reassign" tissue from roots to shoots, so when you have another go with fork, crowbar, etc it comes up much more easily.

I wouldn't start any planting til at least the autumn, depending on what you want to grow, and I'd use the next few months on "bramble patrol" getting rid of them and any other perennial weeds and digging the soil over, adding humus-rich materials and generally getting ready.

In my garden, I had been going to spray any regrowth after several months of this "digging out regrowth" treatment with Roundup, but didn't need to. :-) If you're going to have weeds to clear, brambles are by no means the worst ones you could have.

I use a different method - having cut down as you describe, I then reveal the large, knobbly bit of root under the soil, either drill into it with a small wood bit or make deep cuts, then fill or douse with neat SBK, cover with an upturned pot or something, then replace the soil over that. Takes a bit of time to work, must be left for a few weeks, but as long as you haven't spilt it on the soil, you can get on with other stuff around and about.