It appears that you are concerned about not wanting to hurt your wife's feelings about her current weight gain. This is a good sign that you are mindful of not wanting to hurt another person's feelings. It also appears that perhaps you are still committed to this relationship. A person should always love and be respectful of their spouse, regardless, as to whether or not they have gained weight.
Remember, you married a person, not a number on a scale. I hope that people can love and desire their spouse regardless as to whether or not they have gained weight.
It appears from the comments you have stated in your question, that perhaps this situation may be 'too complex, and/or serious,' for either of you to handle and/or 'fix,' by yourselves, without further in-person professional help.
It appears that you have already realized this by taking the 'first step' by seeking a neutral third party opinion, such as seeking some feedback from 'Just Answer.' This is a good start.
I say this because as we all begin getting older, our physical looks will most likely change. This is a natural progression of life. It is important that you begin addressing this matter now, so you and your wife can enjoy a long and emotionally healthy relationship. Even when people maintain the same weight throughout their entire adult lives, their physical looks will most likely change. We all need to realize that we are not going to look the same way as we did in our youth. (I.E. - As we did during our later teen years, and early twenties.)
Also, perhaps there are some underlying issues which caused your wife to gain the weight in the first place. Maybe there are some 'underlying emotional issues' that both of you are not aware of at this time. This may have been a factor in your wife's weight gain, perhaps not. Maybe there are other issues in 'your relationship,' that are yet to be uncovered. As stated above, it is important to begin addressing these issues now, with the help of professionals.
A good place to start is to contact your 'primary care physician,' or maybe a 'referral service,' at a nearby hospital. Perhaps the 'physician' or 'referral service' would be able to recommend a competent 'mental health professional,' to begin working with you both individually, or together as a couple.
Perhaps these sources could also recommend some in-person 'family counseling,' which would include the kids. (If they are past the infancy stages.)
This situation may be affecting them as well. Maybe some good 'family counseling' could be helpful in addition to any individual, and/or couples counseling, you may seek.
If financial affordability is an issue, then please be certain to mention this when seeking recommendations. This way, maybe they will be able to find competent 'mental health professionals' who would charge more reasonable fees.
Also, perhaps the 'referral service' or the 'physician' would be able to recommend a 'not-for-profit counseling center,' where the two of you, or your entire family, could receive some quality counseling at more affordable fees. Some centers provide an array of counseling services with more reasonable fees, if any at all. This can include, individual, group, family, and marriage counseling, as well as other services. Also, this may include a variety of 'support groups.' (*Every center is different, and the services, and fees, may vary.)
In addition, perhaps your wife could consult with 'her own primary care physician' regarding finding a 'safe weight loss and exercise program' which would help her lose the weight she has gained. This is also important because gaining weight, and/or obesity, could lead to future health problems. You both want to take good care of yourselves, in order to help you each remain physically healthy for each other, and especially for your kids. It is important that a physician is consulted about this immediately.
Some 'weight loss and exercise programs' even have online, or 'over the phone support,' from the programs. This too may vary, as every program is different. Perhaps while you and your wife are consulting the physician about the a mental health referrals, he, or she, could also make recommendations about a sensible weight loss and exercise program as well.
(*As a precautionary measure, before anyone begins a new weight loss, and/or exercise program, they should always consult with their own 'primary care physician' first.)
You mentioned that both you, and your wife, work different shifts and do not share the same days off. If you, and your wife, are allotted any 'paid personal days off,' at work, perhaps you could arrange them on the 'same days' in order to be able to take your days off together. Some companies provide them, some do not. For example, since it appears that you both work different hours, then maybe you both can take several 'paid personal days off,' if your places of work provide these options, and it will not place either of your positions, and/or job securities, in jeopardy.
If you and your wife should need some time together, such as for couples counseling, etc., and need someone to watch the kids, here to are some ideas.
If there is a 'child care issue' involved even if you both have the same days off, perhaps you could hire a baby sitter who will charge a more reasonable fee. If money is tight, maybe a trusted close family friend, or relative, could baby sit for your kids for free.
Also, perhaps there is someone in your family, or amongst your personal friends, who also has children. Perhaps during this time, you or your wife can trade 'baby sitting services' with another couple, or parent. Meaning, that if they watch your kids for free, you and your wife will do the same for theirs.
If everyone is agreeable to this, perhaps a simple 'written contact' could be drawn up regarding the terms of your 'baby sitting' agreement. Maybe you could contact the 'Bar Association' in your area, for a referral. Perhaps they would be able to recommend an attorney who would be able to charge a more reasonable fee, if any at all. Maybe the association would be able to recommend a 'legal clinic,' in the area, where you would be able to work with an attorney regarding the 'baby sitting contract' for a more reasonable fee, if any at all. (*Services and fees by individual lawyers, and/or legal clinics, may vary. You would need to discuss this matter with someone who would have more knowledge on this particular topic.)
As stated above, it is a good start that you have taken your wife's feelings into consideration. It is a positive start that you have not made any 'insensitive comments' about her weight gain, as this would only make the situation worse and could further cause problems in your relationship.
There is a big difference between telling a lie, and simply being supportive and encouraging. In my opinion, it is not dishonest to be mindful of someone else's feelings, and therefore choose to be emotionally encouraging, and supportive.
As I stated above, you and your wife should begin seeking the referrals I mentioned immediately.
I hope the above stated suggestions, and opinions, will be helpful to both you, your wife, and your children.
Good luck!, and best wishes.
Legal Disclaimer: The information provided above, is general information only and is not intended to serve as a long, or short term, professional relationship. By providing the above information, I am not engaging in a 'relationship designee - client relationship' with you. The above information is only intended to provide general information. The fee that you may pay me, is for general information only. No part of this disclaimer can be reproduced, or copied, without the express consent of Rosemary S., the owner.