From what I've seen my coworkers do when they didn't want to talk about some personal thing (like the reason for sick leave) there are 2 main ways to handle any inquiry about it. Which one to choose depends on whether you want people to always take you seriously, or if you want to be known as a bit of a joker outside of serious work related topics.
1) Give ridiculous answers. Make it obvious you're not going to tell the truth.
2) Dodge, dodge, dodge, give something small and thank them for leaving it at that.
I know you asked for solutions that did not involve lying, but that depends on what you see as "a lie". I agree that it feels wrong to say you have the flu when it was mental health issues. But what if you tell an obvious joke instead?
Take this conversation with my coworker for example:
So [coworker], how's the baby doing lately? (he has a 5 month old baby).
Oh man, she has measles, high fever, and the fifth disease*!
(me joking back:) Oh, well at least she's now had those so they shouldn't be coming back .. no wait that's only after they're 6 month old right?
Imus, if she really had all those she wouldn't have survived :p
Since he always jokes in an exaggerated way to personal things he doesn't want to talk about I knew it was pointless to ask for a serious answer. He would've just made up another highly exaggerated story again.
This works as a general approach if you want an easy way to avoid talking about your personal life with coworkers, but might only be a real option for extroverted people overall.
Another coworker is the polar opposite of the joker above. He's the more typical introvert that will avoid talking with most people. If you ask him about his absence he'll probably give you just enough information to see that it's justified, yet not enough to be interesting otherwise.
If he has something like the flu, he'll just say that he has like most people. If it's something different like a surgery he'll only inform a couple of his closest coworkers but dodge most others.
For example: While we knew about what surgery he had on a certain day, his response to someone else asking why he wasn't at work was simply "Oh that was just a doctors appointment. What brings you here?"
Notice how he didn't really say anything at all. Sure a doctors appointment could be serious, but it could also just be a regular checkup. And since he finished with a question on something entirely else, the absence conversation is now over. There's no way for the other person to continue asking why he had to see the doctor without feeling awkward himself, whereas my introverted coworker could just as easily dodge that new question as well.
So more specifically for your situation now. Since you've been gone for a week you're somewhat expected to give at least some information about why you've been gone for so long. Since you don't want to mention mental health issues, your next best thing is to tell about certain symptoms that could be cause by either those mental health issues, or something much more common. You really had those symptoms so you're not lying, yet you'll still be dodging mentioning anything mental health related. After you've given your minimal information you then ask a question leading the conversation somewhere else (or use some other way to shut it down) which makes it really hard for them to come back to it. For example:
So did you have the flu or something?
Oh no it wasn't the flu but I felt really bad. I did barely had any energy left to get out of bed and had trouble keeping my food in. After consulting with my doctor I was told to stay home for a week and rest up. I'm feeling somewhat better now though, thanks for asking :) So how's xyz been going while I was gone?
Note that I meant "doctor" in the general sense, which includes psychiatrists as well.
Replace the symptoms with whatever you were feeling and are comfortable mentioning to your coworkers. Preferably using words that everyone understands from experience (like tiredness, stomach aches, nausea, ...).
If you expect your issues to still be clearly visible you can pre-empt that yourself as well. Just casually mention (after explaining symptoms) that you might still look pretty tired and grumpy for a while and that they don't need to worry about it. That way, the next time you're looking somewhat down, they might just think "oh right, they're still recovering" instead of asking you what's wrong.
You should always end with either a change of subject or completely shutting down the conversation. If you don't you might create an expectation that you should explain more about what you have. A couple of options that usually work after such a week of absence are asking what you've missed while you were gone. Ask for updates on a project you're supposed to work on. Ask about how they're doing. Or just say "... Thanks for asking, now if you'll excuse me I still got a LOT of mails to go through."
*: fifth disease: erythema infectiosum, caused by parvovirus B19.