Spain: The Goya Awards (Premios Goya) are the Spanish equivalent of the Oscars. The first Goyas were awarded for movies released in 1986. The Premios Iris are awarded by the Academia de las Ciencias y las Artes de Televisión; the award for makeup & hair was introduced in 2006. The Catalan Film Academy announces annual Premis Gaudí / Gaudi Awards.
UK: The EE British Academy Film Awards are awarded each February and the BAFTA Television Craft Awards are awarded annually in May. The BAFTA Cymru awards are also announced in May. The Royal Television Society’s RTS Craft & Design Awards are announced in November. The National Association of Screen Make Up Artists and Hairdressers (NASMAH) has in the past announced some awards for makeup and hair design.
Italy: The Premi David di Donatello are awarded by the Accademia del Cinema Italiano and introduced a Best Makeup category in 2008.
Ireland: The Irish Film & Television Academy, a not-for-profit all-Ireland organisation that promotes, encourages and rewards creative excellence in film and television, runs the annual Irish Film & Television Awards (IFTAs).
Germany: The German Film Academy announces the Deutschen Filmpreise winners (the Lolas) in April. The Lola for Best Makeup was introduced in 2010.
Austria: The Österreichischer Filmpreis for best makeup is awarded by the Akademie des Östereichischen Films. The awards were introduced in 2011.
Poland: The Gdynia Film Festival is held each year. I believe the makeup award category was introduced in 2009.
France: the Académie des arts et techniques du cinéma has the César Awards, named after César Baldaccini, but almost alone of the major national film awards they do not have a makeup category.
Denmark: The Danmarks Film Akademi (Danish Film Academy) awards the Robert Arets Film & TV Pris.
Sweden: The Guldbagge Awards (Guldbaggen) are overseen by the Swedish Film Institute and are the Swedish equivalent of the Oscars. The Best Makeup category was introduced in 2012.
Finland: The Finnish Jussi Awards are one of Europe’s oldest film awards, dating back to 1944.The award for Best Makeup (Parhaan Maskeeraussuunnittelun) was first awarded in 1992 but only became an annual award in 2013.
Czech Republic: Czech Film and Television Academy (ČFTA) introduced the Lion Awards in 1994 to recognise accomplishments in both filmmaking and television. There is now an award for Best Makeup but I do not know whether that was first introduced in the restructured 2014 awards or earlier.
The European Film Awards, presented jointly by the European Film Academy and EFA Productions, honour the greatest achievements in European cinema. Beginning with the 29th Awards (December 2016) a makeup & hair category will be introduced.
USA: The Academy of Motion Picture Arts and Sciences announces the winners of the Oscars in February. The Academy of Television Arts & Sciences announces the winners of the 2010 Creative Arts Emmy Awards in September; the Daytime Emmy Awards by the New York–based National Academy of Television Arts and Sciences recognise excellence in American daytime television programming. The Academy of Science Fiction, Fantasy & Horror Films announces the winners of the Annual Saturn Awards in June. Local 706 of IATSE announced five annual Makeup and Hair Stylist Guild Awards but these ceased in 2004; they have been relaunched in 2014. The Broadcast Film Critics Association of the USA and Canada bestows the annual Critics’ Choice Movie Awards. There are also a number of niche & genre awards including the Fangoria Chainsaw Awards.
Canada: The Academy of Canadian Cinema & Television announces the winners of the Annual Canadian Screen Awards, replacing the prix Génie / Genie Awards (film) and the Gemini Awards (television). The Motion Picture Arts & Sciences Foundation of British Columbia announces the Leo Awards. The Canadian Network of Makeup Artists (CNoMA) started what appears to have been a short-lived series of annual awards in 2003.
Mexico: The Academia Mexicana de Artes y Ciencias Cinematográficas announces the annual Ariel Awards.
Argentina: The Premio Sur are the annual awards of the Academia de las Artes y Ciencias Cinematográficas de la Argentina.
Brazil: The Academia Brasileira de Cinema was founded in 2002 and awards the Grande Prêmios each year. The first Grande Prêmio for best makeup was awarded in 2003.
China: The Hong Kong Film Awards Association announces the winners of the Hong Hong Film Awards; the Costume & Makeup Design award winners include costume and production designers as well as makeup artists.
Taiwan: The Golden Horse Awards are awarded each year at the Taipei Golden Horse Film Festival.
India: The Indian government’s Directorate of Film Festivals runs the National Film Awards; a Silver Lotus Award (Rajat Kamal) for Best Makeup Artist was recently introduced. The International Indian Film Academy (IIFA) Awards are presented annually by the International Indian Film Academy.
Australia: The Australian Production Design Guild introduced the APDG Awards in 2011 to celebrate talent in the Australian film, television, stage and interactive multimedia industries.
New Zealand: The New Zealand Film Awards are supported by the New Zealand Film Commission, and naming rights sponsor Rialto Channel. The New Zealand Television Awards ceased in 2013 when TVNZ withdrew support.
The Arabian Cinema Awards were established by Dear Guest Co with the aim of honouring filmmakers in the Arab world in all aspects of filmmaking industry.
South Africa: The SAFTA Awards were established under the auspices of the National Film and Video Foundation of South Africa to honour, celebrate and promote the creativity, quality and excellence of South African Film and Television talent and productions, and to encourage entrepreneurship and the development of new talent within the industry. The makeup categories are discretionary and have varied from year to year.