Okay, first off I think a lot of people have made some really good points about not pushing this too hard. When you're already self conscious, then even well meaning nudges can just make you break down entirely.
One thing I would say though - swimming? Seriously???? I can think of literally no worse sport for someone self conscious about their body!!!! No wonder she is avoiding it!!!
I think three things to try maybe:
Firstly, stop eating junk. Don't even reference her weight or her desire to lose any. Simply couch it as something that YOU want.
So for example, "I'd really like to start eating better in the new year. It would be great if we could set a good example of healthy eating to our kids when they're still young, and to be honest, I'm not that confident in how my cholesterol will fare if we keep on like this. I can't do this alone though, so will you join and support me?"
Then actually HELP with the food shops and the cooking. Compare recipes. Whatever.
Secondly, if at all possible, try and give her 40-45 minutes of "me" time each day. I don't know how feasible that is, but even if it's half an hour. You can tell her it's to give her a break and she can do whatever she wants with the time. She might want to crash and rest. She might want to call a friend. She might want to scream into a pillow. She might want a bath. But, hopefully if you are consistent, she will start using that time at least a few times a week to do some exercise. The important thing is that her break isn't DEPENDENT on her doing exercise.
Thirdly and I don't know really the best way of doing this, but there are some really good workout dvd's on the market, which you - or her - can actually do at home. So you can fit it in without the hassle of getting changed, going somewhere, worrying about if you look okay, feeling like shit next to the size 6 supermodel running at thirty miles an hour without breaking a sweat next to you. If you're tired, you can stop. If you want to do it in your pants, you can.
Two very good ones are Insanity (which is all cardio) and P90X (which is cardio but mainly resistance training). Both are pretty expensive, but you get like 6-8 different dvd's with different workouts on them. I have insanity and really like it.
There are lots of very tacky, very American "motivational" videos on youtube, which are actually pretty inspirational. They are home made and basically chart someone's progress throughout the series.
With a programme like that, you could get it for yourself (they are definitely suitable for both men and women. No zumba involved...), and see if she decides to give it a go as well.
PS - Just to be clear I'm not affiliated with the company that make those videos in any way. I just use one of them, and it's great. I'd like the other one but can't be arsed to get the equipment.