Bloom - Sorry it took so long! Not really sure how much my story helps as I only saw the MFMU a couple of times as I never made the 3rd tri of pregnancy.

2008 - I have congenital heart disease so was immediately a high risk pregnancy. Add on the fact it was a mutiple pregnancy too. So I was referred to MFMU for my morph scan as apparently they thought there was chance of my heart disease causing growth restriction. 22weeks I had another scan and this time a paediatric cardiologist was there and did a thorough check of babies hearts as apparently they considered me to have a 5% chance of passing on a heart defect. babies were fine. 2 weeks later babies were born via c section, however I would have had them vaginally if they were in the correct position, but Twin 1 was breech and twin 2 was transverse.

2011 - Again considered high risk pregnancy, due to CHD but this time I had the added risk of premature birth. I had a cervical stitch placed at 14wks, so had regular (fortnightly) scans to ensure correct placement and cervix length. I didn't bother having this bub's heart checked by a cardiologist this time. Baby was born at 25wks vaginally due to placenta previa.

So to answer your questions:

Did you see them for your whole pregnancy? Or just once? How often? Whole pregnancy, but I didn't get far enough to know if this would have stopped. I highly doubt it though. I am pretty sure I would have been with them for whole pregnancy.
Why did you see them? As above.. It was mainly because I was high risk.. The whole time my babies health was fine (while they were inside)
Did you have a natural birth? Yes, once
Did you return to your previous "model of care" or just see OBs? I just saw OB's

I am assuming that in your case if Bub's heart rate goes back to 'normal' they may feel comfortable with you returning to your normal model of care, but I really am not sure. As I said above, I was referred there as a precaution due to my health, not the health of the babies.