Should I hide the fact that I did a group assignment completely by myself?

Personally I would dodge the issue using

Most of the work is done, but we need to...

I don't think it will really matter which option you go with, but that way you don't have to worry about it.

Anyone setting a group project knows that it is fairly likely someone will end up doing all the work. If they don't take any action to avoid that (and it sound's like they haven't in this case), then presumably they don't really care.

I understand that you want to get the credit for doing the work. But I agree it might come across as tactless to point out the situation in your email. It might be more productive to separately have a conversation with the lecturer about why they haven't taken steps to avoid the situation occurring.

I have been on both sides of the fence.

As a student, I was once part of a trio where the two other members were useless. Not "self-fulfilling prophecy"-useless but "dumb as a rock"-useless.

I did all the work and stated clearly to the TA that it is 100% my work and that they are idiots. I specifically said that I am not looking for special treatment but want to highlight the ownership of the work.

This later (in academic life) translates as "do not add to your paper co-authors who did not do anything, just because you are under the pressure to do so". Surprisingly, the latter is met with nods of approval, while pointing out student leeches is not.

As a teacher, I had twice the case where someone came to me saying that they are the sole author of a two and three-members team. I asked them to highlight how they can persuade me that this is the case. I did not have a difficult situation, in both cases the "real author" knew everything, while the others were clueless. He got the right mark for the exercise (both were good) and the others got zero. I made sure to document the whole story though.

You specifically ask what the lecturer will think, so I'll address that first - though I think that's actually a red herring here.

I have not lectured myself but I have managed people in a business environment, and analogous situations come up: I think the response will depend on the way you raise the issue, and on the lecturer's busyness at the time of receiving your email. It may just appear to them as a dispute between students, and one they don't desperately want to get involved in (this kind of work creates these kind of disputes, and most groups will resolve the problems themselves). So the lecturer may think (justly or not) that you are a bit of a nuisance. Equally they may think you have a just complaint and have sympathy with you.

I think the lecturer's response in this situation will depend heavily on how you have tried to resolve this issue yourself before bringing it to them. If you bring the issue to them without having first discussed it with your group you may (justly or not) look immature and lacking in social skills. Also I think the response you get will depend on how openly you broach it - subtle hints like the one you indicate may just look petty if they are noticed at all: they will probably be ignored: lecturers have heavy workloads and don't tend to seek out issues where they don't need to. An honest and straightforward complaint, backed up by your having made a reasonable attempt to resolve this with the group will get a better reaction.

I don't think the lecturer's response should be your biggest focus here though.

We had a similar situation in a course I recently completed: two out of four of us literally did all the work for a substantial project. My perspective is: don't get hung up on credit, the credit is a tiny portion of the value of doing these things. By engaging you pick up more knowledge and experience, and you will get the benefits of that all the way down your career.

And if the injustice still really bothers you (and honestly it's hard to avoid feeling that way) bear in mind that this is a small portion of the credit required for your course. The knowledge you gained by actually engaging with this assignment will benefit you in other assessed parts of the course where you will work alone. If they get the credit now, they'll almost certainly lose it later.

The ideal thing to do is to be open about your feelings about work sharing with the group, as early as possible. Some social groups make this easier than others though. And in some cases (as in mine) you can have that conversation (and you can have it as strongly as is possible without falling out with them), and still end up doing all the work. Different people want different things out of a course and are willing to put in different levels of effort. Sometimes social pressure overrides that, sometimes it doesn't. That's all you can take away from this really. And you need to honestly judge your ability to have that discussion with them without souring things before you try to have it.

But you will have to work with these people again, and see them again, and you may be in a situation in the future where you could interview for a job at a company where one of them works. Better they remember you as the hard working person than the person who made them feel bad, or "ratted them out" (as they will see it).

So in short - be tactful, there's nothing to be won by doing anything else, and if you need to feel there's some justice in the situation remember that they lose out by not really engaging. That isn't a platitude: 2 years into your career university module credit counts for nothing, knowledge and the ability to work carries you. These people are not accumulating knowledge or learning to work.