Hi hun - wow you have done so well to get to 34wks after your cervix had begun to funnel and you had an emergency stitch placed - AND you are carrying twins!!!! I am really impressed
I had my second daughter at 24wks due to incompetent cervix and my emergency stitch placed at 23wks failed. By the time it was spotted tho, my cervix had already dilated 3cms so the stitch was much less likely to work.
Ok - in my twin pregnancy I had an elective stitch placed at 12wks, and I made it to my planned section at 38wks. When you have a section they don't remove the stitch in advance, they do it after the babies have been delivered whilst you're still in theatre so I can't give you my own personal experience. However, it is usual for ladies with cervical stitches to have them removed, but then labour does not begin for several more weeks.
Despite you already being 1cm dilated, it will take strong, regular contractions for your cervix to dilate enough to deliver two term babies. An Incompetent cervix dilates usually to 3cms 'silently' and without contractions, at 20wks this is enough to allow the membranes and baby to fall out, at 36wks it is nowhere near wide enough lol! Occasionally the stitch removal itself stimulates contractions, but more often baby just sits there, plugs any gap and the body has to go into full-blown labour (usually at or near term) to deliver the baby. Since you are carrying twins, you may labour a little earlier than a singleton lady, but not necessarily, and certainly no earlier than any other twin lady without cervical issues.
Good luck hun, and well done for getting this far x