I did find this in the documentation (https://doc.qt.io/qt-6/qsqlquery.html):

So I moved the query to after if(db.open()), as shown below

QSqlDatabase db = QSqlDatabase::addDatabase("QODBC3");
QString connString = "DBQ=umlaute.mdb;Driver={Microsoft Access Driver (*.mdb,*.accdb)};FIL={MS Access};"
QString sql_statement = "SELECT example FROM test;" //or "SELECT example_utf8 FROM Test_utf8"

db.setDatabaseName(connString);
if (db.open()) {
    QSqlQuery query; //moved query to here
    QFile file("output.txt");
    if(file.open(QFile::WriteOnly)) {
        QTextStream out(&file);
        out.setCodec("windows-1252") // or "UTF-8" if Test_utf8
        query.prepare(sql_statement);
        query.exec();
        while (query.next()) {
            qDebug() << query.value(0).toByteArray();
            out << query.value(0).toByteArray() << " | " << QString::fromUtf8(query.value(0).toByteArray()) << " | " ;                                                    
            out << QString::fromLatin1(query.value(0).toByteArray()) << " | ";
            out << query.value(0).toByteArray().toHex(' ') << Qt::endl;
        }
    }
}
else {
    qDebug() << "could not open database";
    qDebug() << db.lastError();
}

db.close();

Now i get
when querying table test, which is correct.

Thank you for your time and help!

Here's some stuff you could try:
Try using ODBC2 instead of 3:
QSqlDatabase::addDatabase("QODBC");

Try changing the Unicode settings in your Windows PC
https://stackoverflow.com/a/69268839/233402

This is what I get:

and this is what I get when I query table Test_utf8:

I think regarding Test_utf8 the output is as expected.
However, regarding table test the "information" is "lost" upon being read from the database?

adding the following into the while loop:

qDebug() << "output variant: " << query.value(0);

already yields:

so i would need to specify the encoding before the query is executed? How could I do this?
As far as I am aware I can't specify an encoding, e.g. charset="Windows-1252" in the connection string for odbc and ms access.

The actual ms access database I will have to work with will unfortunately be like table test and not like table Test_utf8. I just wanted to see what result I get when I am sure the data in the field is utf-8 encoded.

...
query.exec();
    while (query.next()) {
        qDebug() << QString::fromUtf8(query.value(0).toByteArray());
    }
...

When querying table test I get: 'hell\xEF\xBF\xBD" which is a replacement character and my guess is it indicates an encoding problem.

Those three bytes are the UTF-8 encoding of the Unicode U+FFFD used to replace an incoming character whose value is unknown or unrepresentable in Unicode. My guess is that the data in the column is encoded in Windows-1252, that is a bare 0xF6 byte to represent the 'ö' character.

A single 0xF6 byte does, indeed, make no sense if you try to interpret it as UTF-8.

With table Test_utf8 I get: "hell\xC3\xB6" which corresponds to an "ö"

This is the UTF-8 encoding on the Unicode code point U+00F6 (LATIN SMALL LETTER O WITH DIAERESIS).

In your shot from Access of the UTF-8 column you see these two bytes as "Ã" and "¶" as that is what they represent in the Windows code page.