Global climate change is causing an increasing number of drought events, which might impact the stability of wheat breadmaking quality. In this study, 73 spring wheat lines with diverse genetic backgrounds (modern, old, and wheat–rye introgression) were drought treated, and the grains were analyzed by high-performance liquid chromatography for protein composition traits related to breadmaking quality. The amount of total sodium dodecyl sulfate-extractable and -unextractable proteins (TOTE, which correlates to grain protein content) increased significantly under late drought, while no effect of early drought was found on the analyzed protein composition traits. Under control treatment, genotypes with 3R showed significantly higher TOTE than genotypes with 1R, 1RS, and 2R, indicating the potential role of 3R in increasing grain protein concentration. The lower percentage of sodium dodecyl sulfate-unextractable polymeric protein in the total polymeric protein (%UPP) found in 1R and 1RS genotypes as compared to modern and old genotypes suggested a gluten strength reduction induced by 1R and 1RS. Despite the negative yield–protein correlation found in this study, lines 252 (3R), 253 (3R), and 258 (2R) displayed the presence of germplasm with both high yield and protein concentration. The %UPP was found to be positively correlated to spike-size-related traits (grains per spike, grain weight per spike, and spike length) across all three treatments. Additionally, high and stable TOTE was mainly obtained in genotypes with 3R, while old genotypes showed dominant performance in %UPP. Thus, genes responsible for high and stable protein concentration and gluten strength should be explicitly searched among introgression lines with chromosome 3R and old Swedish cultivars, respectively.