Get each individual EU-DEM tiles from http://www.eea.europa.eu with a batch. You need wget (apt-get install wget for Ubuntu/Debian or on Windows go to http://gnuwin32.sourceforge.net/packages/wget.htm for installing) After, just do in command-line wget -c -i dem_eu.txt

dem_eu

http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_2500015000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_2500025000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_2500035000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_2500045000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_2500055000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_3500015000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_7500015000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_7500025000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_3500025000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_3500035000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_3500045000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_3500055000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_4500015000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_4500025000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_4500035000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_4500045000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_4500055000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_5500015000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_5500025000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_5500035000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_5500045000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_5500055000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_6500015000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_6500025000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_6500035000-AA.tif
http://published-files.eea.europa.eu/eudem/jrc_r_3035_25_m_gsgrda-eudem-dem-europe_2012_rev1/eudem_tiles_1000km/EUD_CP-DEMS_6500045000-AA.tif