Have there been any attempts made to make a comprehensive list of all scrolls or pieces of scrolls of the New Testament (in original languages) that have ever been discovered by archeologists?
For the New Testament, yes. Though Wikipedia may not be a "scholarly" source I find it good for this sort of thing.
For the Greek Old Testament (i.e., "Septuagint"), there is also the Verzeichnis der griechischen Handschriften des Alten Testaments [= Catalogue of the Greek Manuscripts of the Old Testament], originally compiled by Alfred Rahlfs and published in 1914. It is available from Archive.org, or in a hi-res PDF from the Göttingen Septuaginta-Unternehmen.
It was revised in 2004 by Detlef Fraenkel -- see the SBL bookreview for a description of the work, and the changes from Rahlfs' original edition.
[I know OP is interested in New Testament manuscripts, but this catalogue has many manuscripts that also contain NT material, and comments on the question suggest a slightly wider interest beyond the limits of the question as posed.]