Gonzo- you can't fly the original GA as written. You are looking for a DME reading off the ILS to make your turn north or south as appropriate to make your turn away from the airport at LHR. If you have moved to the other runway, you are not going to get that DME reading at all, so it is impractical. If what you say is correct, then the whole thing is dangerous as if you fly the original GA, you will be turning right over the central area to cross over the other runway- a bit crazy considering that somebody may have been cleared for take-off on it. When I have done this procedure in the past at LHR, we have had time to retune the ILS and get the GA read out. So when you GA, where does that leave you.
It is largely a theoretical discussion anyway as BA issued a notice to refuse swingovers like this a few years back. The reason was we are to be stabilised at 1000' at the latest lined up with the runway. Therefore you have to be changed over much further back, which makes it a whole new approach, with time to appraise yourself of the procedures for the new runway.
I now know enough not to accept them at all, as ATC procedures appear to be so confused, unless it is quite understood what the GA procedure is to be. Because of the doubts about it being thought through enough at LHR at least, that is most certainly a place to decline them unless it is agreed what must be done. Discussing potential GAs below 1000' is daft. Don't do them then. Especially when company procedures ban them.