Anything that has glyphosate as the active ingredient should be OK, BUT it only works when the plant is actively growing, if you're in the south you should still be OK (just), but if elsewhere you will need to wait until spring to use it. Roundup is OK, I tend to buy the sachets of extra-strong glyphosate from ebay, each sachet makes up one litre, you can get a two-litre sprayer that's re-usable from Amazon for about 8 quid.
You have to spray the leaves of the unwanted plant, so if there are roots around with no leaves, you would have to wait until the thing sprouted up and then squirt it. It will work but it will take repeated applications for things like brambles, eventually the plant will be so weak, it won't have the energy to re-grow. Unless you are in the far south you will be wasting money, as up here in Yorkshire things are starting to die back, and it won't be long before we have the first frost of the Autumn.
My advice would be to dig up as much as you can manage now, bit by bit, and then start with the weedkiller once things start to shoot up in the spring. You may want to wait a bit before investing in any plants, especially bedding plants, as you don't want them to have to compete with the weeds. I hate Ivy, but I hate brambles more, as at least Ivy doesn't have thorns!
If you're like me and really strapped for cash, wait until your missus has finished with a bottle of cleaner (the one with the trigger-type spray, somthing like dettol or kitchen cleaner), rinse it out really, really well, get an empty two-litre bottle, rinse, get two sachets of Glyphosate from ebay (costs about £4), and make up the two-litre bottle. You'll have to keep re-filling the squirty trigger bottle, but for £4 and a bit of faffing and re-cycling you have a powerful weedkiller that's as good as anything branded. The weed has to be GROWING or it won't work, you'll just be chucking money away.
Typical Yorkshirewoman, short arms, deep pockets