While it is well established that societal restrictions have been effective in reducing COVID-19 emergency demand, evidence also suggests an impact upon emergency demand not directly related to COVID-19 infection. Hospital planning may benefit from a greater understanding of this association and the ability to reliably forecast future levels of non-COVID-19 demand. Activity data for Accident and Emergency (A&E) attendances and emergency admissions were sourced for all hospitals within the Bristol, North Somerset and South Gloucestershire healthcare system. These were regressed upon publicly available mobility data obtained from Google's Community Mobility Reports for the local area. Seasonal trends were controlled for using time series decomposition. The models were used to predict non-COVID-19 emergency demand under the UK Government's plan to sequentially lift all restrictions by 21 June 2021, in addition to three alternative hypothetical relaxation strategies. Rates of public mobility within the local area were shown to account for 77% and 65% of the variance in non-COVID-19 related A&E attendances and emergency admissions respectively. Modelling supports an increase in emergency demand in line with the level and timing of societal restrictions, with significant increases to be expected upon the ending of all legal limits. This study finds that non-COVID-19 emergency demand associates with the level of societal restrictions, with rates of public mobility representing a key determinant. Through predictive modelling, healthcare systems can improve their demand forecasting in effectively managing hospital capacity.