Not sure how it will work out with this little guy but with my daughter I was back to prepregnancy weight in 6 weeks and a few months after birth I was at the lowest weight of my adult life. I was actually at an ideal weight, then I put on 56lbs over the course of the next 7 years - nobody to blame for being fat but me.
I am pretty confident that the weight will just fall off again this time but breastfeeding helps with that a lot.
1200 is way too low for just having a baby, breastfeeding or not! You need plenty of nutrients for your healing body, protein especially. And, iron for blood loss.

With my first I didn't count calories. I was 22 lbs overweight, I had gestational diabetes in 3rd tri and lost 6 lbs in 1st tri. On my GD diet I ended up only gaining 23 lbs going overdue (the last 6 lbs were weeks 39-41). By 8 days post partum I'd lost the 23 lbs. When my son was 6 months old I joined a gym and lost another 10 lbs before getting pregnant again at 9 months. Still not counting calories. I ate EVERYTHING. He didn't really eat solid food until he was 9/10 months old, much preferred nursing.

My second I was convinced I'd have GD again, so ate everything thinking I would need to diet again 3rd tri. Didn't get diabetes, but I did pack on 46 lbs. By the time I weighed myself 2 months post partum I'd lost 22 lbs. For the next 5 months I counted calories - usually 1600-1800 as I was nursing and HUNGRY. Lost the other 24 lbs and another 12 lbs and then got pregnant again! (Planned, we wanted them close together, didn't realize they'd be THAT close together though). So by 7 months post partum (and 1 month pregnant) I was down 58 lbs.

Started this pregnancy at a NORMAL/HEALTHY weight! Woo! Been eating 1800ish calories (I'm only 5'2"/5'3") but definitely more on days I want it. I've gained 20-22lbs so far (I've lost 2 lbs this week?! - hopefully that means birth is close, either way it's weird for me as I normally gain 1-4 lbs a week at this point being 38 weeks!). I'm HOPING most of it will come off relatively easy/quickly since I seem to lose about 22-23 lbs from birth (but I was more overweight to start with then so it was probably easier).

I'll eat more calories breastfeeding than pregnant (looking forward to it!!!) as you burn like 500 calories BFing vs needing an extra 200-300 pregnant. MORE FOOD. Do not cut your calories too much, your baby may get unhappy! You're supposed to be extra careful the first two months to establish a good supply.

Edited to add with my second to lose the weight all I did was eat reasonable and walk. I threw both kids in a double stroller (or when the youngest was little a pack and the older one in a stroller) and walked to the park and back (1.5 miles if I took the long route). So it might have come off even faster if I had worked out for real instead of walking.

Has anyone had any luck losing their weight fairly quickly after the baby comes? And how long did it take? What was your calorie goal during AND after the pregnancy??

Right now, I'm at 2,000-2,5000 a day, but I plan to cut it back to no more than 1,200 a day once my little guy is here, and I would like to work out ASAP.

Just getting anxious to get my body back now that I'm a couple days past 36 weeks! )

A lot of it falls off in the weeks after pregnancy. I agree with the others, 1200 is too low right off the bat. Your body needs a chance to heal from everything it's been through in the last 9 months! It takes you almost a year to put it on, give yourself a year to take it off.

If you're breastfeeding, that will help as long as you're eating healthy. I lost a lot of my weight while nursing... but once my son started sleeping through the night and cut out about half his nursing sessions simultaneously, I didn't adjust my calorie intake down... and I gained a lot of it back. So keep that in mind as well. I never had an issue with my milk supply dropping when I cut back calories, but I had a major issue with it when I didn't drink enough water! So if you do plan to nurse, every time you sit down to feed the baby plan to drink at least 8 oz of water (or more, if you can).

But above all, think about the amazing things your body accomplished over the last 9 months... you grew a human being! Allow yourself some time to heal and adjust. I'm assuming this is your first? Your first couple months, don't even try to focus on losing the weight. A lot will come off naturally, especially if you eat healthy. Your biggest focus is going to be adjusting to a new life with the baby! After a few months and you've settled into a routine it'll be easier!