Increasing temperatures, heat waves, and reduction of annual precipitation are all the expressions of climate change (CC), strongly affecting bread wheat (Triticum aestivum L.) grain yield in Southern Europe. Being temperature the major driving force of plants’ phenological development, these variations also have effects on wheat phenology, with possible consequences on grain quality, and gluten protein accumulation. Here, through a case study in the Bolognese Plain (North of Italy), we assessed the effects of CC in the area, the impacts on bread wheat phenological development, and the consequences on grain gluten quality. The increasing trend in mean annual air temperature in the area since 1952 was significant, with a breakpoint identified in 1989, rising from 12.7 to 14.1°C, accompanied by the signals of increasing aridity, i.e., increase in water table depth. Bread wheat phenological development was compared in two 15-year periods before and after the breakpoint, i.e., 1952–1966 (past period), and 2006–2020 (present period), the latest characterized by aridity and increased temperatures. A significant shortening of the chronological time necessary to reach the main phenological phases was observed for the present period compared to the past period, finally shortening the whole life cycle. This reduction, as well as the higher temperature regime, affected gluten accumulation during the grain-filling process, as emerged analyzing gluten composition in grain samples of the same variety harvested in the area both before and after the breakpoint in temperature. In particular, the proportion of gluten polymers (i.e., gliadins, high and low molecular weight glutenins, and their ratio) showed a strong and significant correlation with cumulative growing degree days (CGDDs) accumulated during the grain filling. Higher CGDD values during the period, typical of CC in Southern Europe, accounting for higher temperature and faster grain filling, correlated with gliadins, high molecular weight glutenins, and their proportion with low molecular weight glutenins. In summary, herein reported, data might contribute to assessing the effects of CC on wheat phenology and quality, representing a tool for both predictive purposes and decision supporting systems for farmers, as well as can guide future breeding choices for varietal innovation.