I hope I put this in the correct forum.

I have a moral issue I am trying to get over.

I have a great job at the moment working on a project I really enjoy but was just offered another job somewhere else and the compensation is much better.

The moral issue comes in where I just committed to a 3-month long project at work and I am the only Engineer on the project. The only other Engineer at the company suitable to replace me is going on vacation next week for 3 weeks. If I take the new job and start right away this project will not be able to be finished in the current timeline. I do not want to put my company in this position.

The reasons I am considering the new job;

- My current job is moving 45min farther away then it is now in 4 weeks. ( means an extra 45min commute + transit fare. ) I walk to the current office. This adds 2 hours to my day
- The pay is 12k+ more a year plus a yearly bonus 
- The company has better health benefits + Ill get 2 extra weeks of vacation a year
- My wife just went off maternity and is going back to work ( we now have added expense of daycare and even more because of how far the new office is )
- I'm not sure if my current company will be here next year or at least the office I work with ( hiring has shifted to other offices lately )
- I was planning on switching companies after the current project because of the office location change ( but this offer came up before I started looking )

So if I stay to finish the project, the new company cant wait that long but I really do not want to put my current company in that position. But I also realize that if the roles were reversed the company would most likely have no issue laying me off if the need arose.

It sounds like you have several good reasons to accept the new job.  It's up to you to consider what "committed to a 3-month long project" means, as in what type of commitment you made.  You dont tell us.  So, it's your call as to whether you're breaking some moral code of yours.

But generally I'll say that if the company put all their eggs in one basket, so to speak, by having a new project that relies so heavily on just one person, then they are the ones who put the "company in that position" and not you.  But again, I dont know what commitment you made to them.  Maybe you just accepted the project, or maybe you gave your word that you would be there until completion.  Those are different levels of commitment.

Assuming your "commitment" was just to accept the project, then I'd say you should take the new job because of all the reasons you gave.  You need to do what's best for you and your family, and let the company do what's best for them.  If the tables were turned and the company thought it would be best to let you go, they might feel very badly about it, but at the end of the day I'd expect that they'd do what's best for the company.

You can also offer to help them with the transition to a new engineer on the project.  I'm not sure what the specific details of that would be, but maybe it's something to consider.  Maybe you could even work for them on a freelance basis for a while to help them transition, and ask the new company if they'd allow you to work there part-time while you do that.  I dont know, you have to think about it.  But there might be some way to mitigate the transition for your old company if you really want to.

You didn't say how far you are into the 3 months.

A good way of making a decision like this is to weigh the pros and cons using a decision grid.

It is nice to be able to give a two week notice, but other than that your decision should be made considering what is better for you and for your family. With the company you work for it's a business relationship and no more than that. It is their responsibility to ensure that they don't depend on one person, and if they do, is their responsibility to do whatever is needed to keep that person. As long as you don't breach a contract there is no moral issue here.

If your only issue is feeling some guilt, leave it aside and make the decision that is best for you and your family. Rest assured that the other side will do the same.

During the interview for the job I have now, I told the interviewer that I felt it was my obligation to help my previous (then current) company finish an important project and asked them to let me start 2 months later.

I expected that to be a problem for them, but quite the contrary: After the interview, they told me they had talked about that and found it meant I was a loyal person and that they felt assured that if I ever left the new company, I'd give them enough time to find a replacement, too.

Have you tried talking to the new company about the issue? Maybe they are more open to it than you think.

I don't envy you your position but it's good that you're trying to do the right thing for everyone. I'd do what costin and rnlf_in_space suggested. Talk to both companies about a transitional period and try to take it.

On a totally off-topic note, in many countries in Central Europe you wouldn't have such a problem, really, because the notice period is 2 months starting with the following calendar month, for both sides (i.e. 2-3 calendar months effectively, a longer one is negotiable, a shorter is illegal). 

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Maybe you just accepted the project,

Yeah, this is the commitment I made, but because I said I was able to do this they pushed it ahead of another project.

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You didn't say how far you are into the 3 months.

4 days in
 

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If your only issue is feeling some guilt

This is my only issue other than the new job will not be in games but I can live with that for the time being.
 

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Have you tried talking to the new company about the issue? Maybe they are more open to it than you think.

Yes, this has been discussed and the latest I can start is after the first week in Feb.
 

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the notice period is 2 months starting with the following calendar month,

Wow had no idea, 2 weeks is the standard here but I have seen companies negotiate up to a months notice before on both ends of the deal.

In the end, you must do what's best for you. But, you must also be careful to not burn a bridge. If you do decide to leave, be open and honest with your former team and let them know why you're leaving. And if you can, ask the new job to give you a week or two to wrap things up with them so that you can move on without guilt.

You might also consider taking the offer to your current employer, and ask them to provide a counter offer to be competitive with the 12k raise + bonus + commuter benefit.

Hard to predict whether or not they will counter, without knowing the company. But it's largely a win-win situation: if they do counter, you may have the option to stay, and if they don't, you get to walk away with the knowledge that you have been transparent about your reasons for taking the other job.

I second what @swiftcoder said but also that 4 days in the project is pretty much nothing. Also the fact that they are moving places whilst so early in the project means that what you originally agreed to has changed and it's all the more reason to let them know that you can't continue this.