It probably depends whether the DME is only associated with the ILS. If so, and the way to tell is that the range is usually zeroed to the threshold, without the ILS there is no use for it. Well that's the technical answer anyway.
With no use, it will probably be switched off too, maybe automatically, much as every type of kit I've worked with will not radiate a glidepath if the localiser fails unless an engineer manually switches things on.
Hope this helps.