40x30x15cm underseat bag

If the bag is not stuffed to the gills, then it will fit. I don't look for exact specs but instead look for something that will fit within the stated dimensions.
You could also rely on the overhead bin, but place a smaller bag within that you can pull out and keep the things you want at hand under the seat. It is nice to not take up all of your leg room with stuff.

I don't usually reply to these as it can obviously appear self-serving, but as it sounds like you've had some trouble, we do have a few bags that fit that size (or if a dimension just barely exceeds it, they're soft sided so that they can fit requirements... just don't overstuff the bag).

Check out the Oslo and Veloce bags here: https://store.ricksteves.com/shop/travel-bags/day-packs

Or this one: https://store.ricksteves.com/shop/p/flight-bag

I'm sure competitors have other good bags too. Don't let my RS badge on my posts give any sense of superiority over other replies. I'm just the moderator. ;)

I use a Fjalraven Kanken 15" laptop bag. Its dimensions are 40x28x16, so 1 cm too deep. I just don't stuff it full.

You might also take a look at the Rangeland Lightweight Under-seat Backpack available from Amazon. I have one of these and have found its rectangular shape much easier to pack than any of my other underseat personal items.

Its depth is larger than your listed 15cm, but it has straps inside and outside to cinch it down. I've never needed the extra cm. Like others have said, don't fill it full.

Edited to add: These come in great colors, are very reasonably priced, have a strap to go over U-shaped roller bag handles and are lightweight. Here's the link to the Rangeland New Business Trip Backpack 21L Flight Approved Carry on Daypack on Amazon. Note that there are other Rangeland bags similar to those recommended by others.

When I couldn't find one for Icelandair, I made a zippered tote to the exact measurements. Perhaps a bit extreme for some, but it fit in the sizer (with not a lot of room to spare).

I used this one on Icelandair with no problem, despite it being just slightly larger (1/2") all around. As other have said if it's not packed full it should fit in the sizer. The problem is if its weight is combined with your carryon's- but you didn't mention that?
https://us.hedgren.com/products/camden-sustainably-made-tote-blackhttps://us.hedgren.com/products/camden-sustainably-made-tote-black
Not sure about gender or preferences. I do like its features, which doesn't include a ton of little pockets.
Drawback: it doesn't hold a soft wrap for long flights, but I just wear that and can roll it up and semi-stuff it on top during the flight.

Trying to find this as well, and I have yet to find one. Seems crazy that I am flying British Air Business Class and I am nervous about finding this bag.

Remember, a soft sided bag is not rigid. It doesn't fill itself up to those dimensions--especially the thickness. The dimensions given are usually the max they can be. So if you only put in 5 inches of stuff into a 7 inch bag, it will only be about 5 inches. It won't insist on puffing itself out to 7 inches.

What is the size of your "almost" small enough bag?

Regarding flying BA in business, I've done it numerous times and no one has given a second look to my personal item.

We recently flew British Airways business class flight and I used this bag recommended by one of the forums expert packing enablers, Jean perhaps :)

https://www.amazon.com/Sherpani-Soleil-Travel-Backpack-Crossbody/dp/B07P12RDF5

The bag worked perfectly for under seat storage. I did carefully cut off the backpack straps as I will never use them and I found a discontinued color on sale.

With the dimensions you listed - look at backpacks or tote bags or over-shoulder bags.
Is this bag going to be your only bag or a bag to carry flight comforts and a few “just in case” items? Even underseat rolling cases most likely will not fit these dimensions. You need something that can compress.

LLbean has a medium zip-top canvas tote at a reasonable price. For big bucks - there is the Tom Bihn cadet. (Laptop business bags). These are about 17 liters in size.

I bought this bag for my upcoming trip and really like it. It gets good reviews on Amazon and the 16” one (sells out frequently but comes back into stock fairly quickly) meets European airline underseat size requirements.

ECOHUB Travel Backpack 16'' Personal Item Backpack
https://a.co/d/00TrYX0Y

If you want a dedicated laptop pocket, this version has that (takes away space from main clothes section).
https://a.co/d/0aJOk8CD

This is my under seat bag...

https://www.eddiebauer.com/p/82300168/stowaway-packable-20l-backpack?sp=1&color=Sprig&size=ONE%20SIZE&sizeType=

just do not stuff it full.

I use either my regular hiking daypack or a LeSportsac bag for this.

I have the Ecohub backpack with 'laptop" compartment mentioned above.

I don't use the laptop compartment for a laptop. I store papers, notebook, pamphlets, etc. If empty, it lies pretty flat.

Regarding the thickness of 20 cm.....the bag has external cinch straps so if you don't pack it too full, you can cinch the straps to lessen the thickness of the bag.

The only thing I don't like is it has too many compartments which I think takes away space. I'm going back to the Ecohub holdall.

I, too, use the Sherpani Soleil crossbody/tote/backpack on BA with the above measurements. It's perfect, IMO, and have been using it for several years. I use the backpack straps sometimes, but find the crossbody strap the best for when I'm in a hurry to get moving. I use the tote handles for piggy backing it onto my wheelie.

Lots of suggestions, but maybe what I missed in the conversation is: What airline are you flying that you are getting these dimensions? Or are you just looking for a general use bag that fits lots of airlines? Also, what is the purpose for the bag? As a personal item? Or as your primary bag?

I guess my general thoughts are, for a personal item intended to go under the seat, as long as it looks like it will fit, they worry less about specific dimensions and weight (broadly speaking for airlines). I have a bookbag type backpack, that I have never measured, but likely exceeds those dimension, and have never had an issue or a second look on US airlines as well as European, including Ryanair and Easyjet. Of course it is not stuffed to the gills, and easily fits under a seat

Have you considered looking at luggage geared toward children?

Put your feet on the floor in front of you. Or, if your bag sticks out a bit - rest feet on top of your bag. Problem for tall, long legged people. Not so bad for us shorties.

It's a bit late since the original post, but I have been doing some research on this for an upcoming Lufthansa flight, where the dimensions for a personal item are even more restrictive. For a 40x30x15cm item, the Patagonia Atom Tote works (just 1cm over in the height, but as a soft bag, it would likely fit a sizer if not stuffed), the Sherpani Soleil Convertible Tote, or the Curso Nylon Pack. Just to offer some ideas! There are a lot of smaller items that fit as well, but I have been focusing on those that maximize the allowed space. The Timbuk2 Scholar convertible tote is one that fits the even more restrictive Lufthansa depth of 10cm, and I have one on order that I'm going to test out.