In theory, yes, because it works 'through the green', in other words, it should be applied to leafy, vigorous growth, when the ground is moist or the plants are at least not in drought, from where it gets transported back to the root.

However, there have been studies done which suggest that Roundup in particular (rather than glyphosate on its own) gets into root crops such as carrots, though not necessarily enough to stop them growing. I'd also add that neither Roundup nor glyphosate are likely to actually kill ground elder completely.

The key is to spray just enough to wet the leaves of the ground elder, with little or none touching the ground. Mixing a surfactant into the RoundUp will help with that, since you will need to spray less to cover the leaves. As Bamboo says, at minimum, it will take several applications, and ground elder may never go away. Be careful handling RoundUp, since it is more toxic than we used to think.

Be careful with aerosol drift

Interesting answers, which surfactant would you suggest would you suggest Tug and in what quantity?

Round up already has surfactants of various sorts in the mix which is what makes it more of a problem, but I'd recommend using straight glyphosate, not Round up, and add a squirt of washing up liquid to the mix as a surfactant instead. I'll be interested to see Tug's response though...

Bamboo, the last time I used RoundUp itself was about 15 years ago, because they started using additives that made it less effective on perennial weeds. It certainly didn't act like it had surfactants back then.
Jpeter, the surfactant that I use most often, and I'm most familiar with, is Hi Yield Spreader Sticker, and the active ingredients are alcohol ethoxylate and alkylphenol ethoxylate. Other brands of non-ionic surfactant will probably be just as good, though. As Bamboo says, washing up liquid will also work as a surfactant, though it destabilizes the ready to use glyphosphate products, and whatever you don't spray, you might as well throw away. That's less of a problem when mixing it from concentrate, since the mixed spray rarely lasts more than 12 hours, anyway. Yucca extract makes an excellent surfactant, but I don't know how it acts in terms of destabilizing the RTU's.
As for how much: just follow the directions on the bottle.

Roundup comes with a new two way nozzle. Turn it from off and you have a stream; turn it the other way and you have foam. You should have this in the UK by now. The old nozzle had stream and spray. I use foam for more precise applications. I also cut the bottom out of a 8 inch and 6 inch plastic flower pots which I use as an isolating collar. If I am in a tight squeeze with regards to other nearby plants I place the collar right over what I want to eliminate and apply. This way I am assured that I will not miss the weed or splatter on plants close by. It works quite well actually. I don't know if this will help you with plants that pop up through the now unwanted ground cover though. Expect some unwanted collateral damage since I think that roundup might transfer itself across intertwining root hair membranes from different plants. Just a suspicion on my part from what I have seen in my garden and I have seen no formal studies on this.

I should have said glyphosate concentrate, not ready mixed sprays ...