In situations like this the DVD will usually contain more than two subtitle tracks for English.
There will be the fully subtitled track of all dialog and one called 'forced' subtitles that will turn itself on to give you subs for non-English dialog.
You have a couple of different options here (most aren't fun). If you haven't altered the runtime at all, you can extract the forced subtitles and use them as they are.
If you've changed the length of the program you need to use OCR software to generate a text version of the subtitles. With subtitle editing software you can retime it to match your new edit, and then generate new graphic (DVD format) subtitles.
The only other answer is using something like avisynth with a vobsub filter and 'burn in' the subtitles while re-encoding the video. You can then use the re-encoded version in your edit.
Anyone have any other suggestions?