Tick-borne diseases are among the challenges associated with warming climate. Many studies predict, and already note, expansion of ticks' habitats to the north, bringing previously non-endemic diseases, such as borreliosis and encephalitis, to the new areas. In addition, higher temperatures accelerate phases of ticks' development in areas where ticks have established populations. Earlier works have shown that meteorological parameters, such as temperature and humidity influence ticks' survival and define their areas of habitat. Here, we study the link between climatic parameters and tick-related hospital visits as well as borreliosis incidence rates focusing on European Russia. We have used yearly incidence rates of borreliosis spanning a period of 20 years (1997-2016) and weekly tick-related hospital visits spanning two years (2018-2019). We identify regions in Russia characterized by similar dynamics of incidence rates and dominating tick species. For each cluster, we find a set of climatic parameters that are significantly correlated with the incidence rates, though a linear regression approach using exclusively climatic parameters to incidence prediction was less than 50% effective. On a weekly timescale, we find correlations of different climatic parameters with hospital visits. Finally, we trained two long short-term memory neural network models to project the tick-related hospital visits until the end of the century, under the RCP8.5 climate scenario, and present our findings in the evolution of the tick season length for different regions in Russia. Our results show that the regions with an expected increase in both tick season length and borreliosis incidence rates are located in the southern forested areas of European Russia. Oppositely, our projections suggest no prolongation of the tick season length in the northern areas with already established tick population.