Hmm, well a few things to say. First, if you want to use a weedkiller on anything, buy glyphosate, not Round Up. Round up is glyphosate, but it has other additives which make it more ecologically unfriendly than glyphosate on its own.
Second, oh, if only spraying with glyphosate would actually kill blackberries (brambles) and ivy. It won't, it'll barely shrivel some of the leaves. The only answer with both of these is to cut them down and either dig out the roots, or treat by cutting into them and using SBK, a brushwood killer, cover, and leave it to do its work.

As for the best time, clearing the brambles right now would be good, and by that I mean cut them down, to prevent any hedgehogs which may be about deciding to hibernate in there somewhere. Nesting is over for the year, so all you have to think about now is various creatures looking for hibernation places, so now's a good time.

Ivy doesn't usually kill trees unless its really thick. Its easy to discourage it without killing it completely by simply cutting through the stems near the ground. The top growth slowly dies off then and after some months you can just pull it off. (The thicker pieces make wonderful firelighters if you know anybody with an open fire) It will grow back eventually but takes quite a long time. It flowers in winter on the mature stems and is a very valuable source of pollen for some insects flying then, so if you care about wildlife don't remove all of it.

Unfortunately, the ivy we have is invasive, very thick and very quick-growing. It has already killed two trees in the front garden of my house (one of which started to fall over and had to be cut down). It has also killed HUNDREDS of trees in this area of North London (Muswell Hill N10). I wrote to Haringey Council's Environmental Control Dept two years ago but they said the problem is too big for them to deal with on public property, as it grows right back the next year -- all the way up the tree -- and it would be a continuous, year-round job that would never end. It DOES kill the trees -- big oaks, chestnuts, maples, etc. There are dozens and dozens of such dead trees with nothing but ivy covering them up to 20-30 ft and more.

Only private properties with expensive gardening companies stay on top of it -- they cut a good one-foot gap in the ivy at the bottom of the tree EVERY YEAR.

I did what you advise in April 2011 four of the very large trees in this wild area, but the ivy grew back the next Spring and is now all the way to the tops of these trees. Something like Round-up HAS to be used, because spraying the leaves also kills the roots.

On the local public walk/bike way called Park Walk, which is made along a disused railway and is 4 miles long between Muswell Hill N10 and Finsbury Park N4, you can see dozens and dozens of trees in this condition. Ivy and brambles seem to be taking over the wooded areas completely in many places. Along the main road here, Colney Hatch Lane N10 from numbers 10-30 and see several huge trees in expensive gardens already choked with ivy. You can even see this on Google Earth.

I can send you photos if you like.

I agree that a thick, choking, covering of ivy on trees will eventually kill a healthy tree. But, as I said, if you think using Round Up will kill the ivy.... unless you're intending to use their root and stump killer rather than the spray intended for green growth, it won't. Although you're very welcome to try, but do bear in mind it is illegal to use any herbicide/pesticide on public ground.