Hugo Engel was the son of Jacob Engel, born in 1857, and his wife Therese, née Schaffer (1855-1912). Jacob Engel owned a jewellery shop at Große Mohrengasse 20 in Vienna’s 2nd district. In 1912, Hugo Engel registered an ‘antiques trade with art objects’ with the trade authority and the commercial court at Wiesingerstraße 3 in Vienna’s 1st district , but soon moved the business to Zirkusgasse 3 in Vienna’s 2nd district. In the same year, he married Ethel, née Terry-Newmon, a Protestant born in London in 1886, who had converted to Judaism, in the Vienna City Temple. Their daughter Dorothee (Dorothy) was born in 1912 and their son Herbert in 1916. In 1918, Hugo Engel resigned his trade licence and had the company dissolved. After graduating from a painting academy in Florence, he ran a trade in jewellery and clocks at Zirkusgasse 3 until 1924, and from 1928 a shop for oil paintings by old and new masters (Hugo Engel, Tableaux, Vienna), which exclusively supplied other dealers and was geared towards exports. Although Engel spent most of the following years in Paris, he continued to conduct many of his transactions through his main Viennese business.

Hugo Engel fled to France on 20 March 1938, shortly after the annexation of Austria to the National Socialist German Reich. According to the declaration of assets, his total assets, which also included a villa in Weidlingau, totalled around RM 15,600 . The provisional administrator of the business, his former chief restorer Wilhelm Kainz, terminated the tenancy at the end of September. Between January 1938 and February 1939, the Central Office for the Protection of Monuments received more than 60 export applications under the name Hugo Engel, many of which were re-exports. At the end of 1938, Kainz sold 17 old musical instruments from Engel's private collection to the Rück brothers in Nuremberg for RM 300. In February 1939, the Property Transaction Office transferred the liquidation of Engel's company to the Kommerzialrat (Commercial Councillor) Otto Faltis. Faltis exported the entire remaining stock of goods to Stockholm in the ‘second sale to Sweden’ in July 1939.

In July 1938, Engel opened the art dealership Hugo Engel (France) with Etienne Bisson as partners on Boulevard Malesherbes in the 8th arrondissement of Paris. After the beginning of the Second World War, he was initially interned by the French and later by the German occupying forces. Although he had transferred all his shares in the business to Bisson by May 1941, the latter merely acted as his front man. In the same year, Engel began working closely with Karl Haberstock, who was also active in France on behalf of Hans Posse, Hitler's special representative for the planned ‘Führer Museum’ in Linz. Engel traded in works owned by Jewish refugees, such as the Viennese industrial magnate Richard Neumann, and did business with the German art dealer Hildebrand Gurlitt. As the Germans benefited from Engel's expertise in procuring important works of art, the military commander exempted him from the obligation to wear the ‘Jewish star’ in 1942. Engel's son Herbert, who had been living in France since March 1937, also worked for Haberstock, albeit in the southern zone, which was unoccupied until November 1942. He arrived in Switzerland illegally in 1943. Hugo Engel fled Paris in February 1944 to escape the threat of deportation and arrived in Switzerland in April. Because of his age, he was exempted from compulsory labour and was allowed to move in with his daughter Dorothy and son-in-law Fritz Feierabend in Zurich as a civilian internee. The assets of Hugo and Ethel Terry Engel and their children that were still in Reich territory were forfeited to the German Reich on the basis of the 11th Ordinance to the Reich Citizenship Act of 25 November 1941. Hugo Engel's father Jacob Engel, who had also fled to France, was deported from the Drancy transit camp to Auschwitz in February 1944 and was subsequently murdered.

Hugo Engel returned to Paris in September 1945 and soon became managing director of the art dealership Hugo Engel (France) again, which he relocated to Boulevard Courcelles. In 1947, the IXe Comité de confiscation des profits illicites de la Seine opened an investigation into Engel's dealings with the Germans during the Second World War. As it assumed that Engel had not acted under duress, it confiscated 822,000 francs and sentenced Engel to pay a fine of 2.4 million francs for tax evasion.

Engel's villa in Weidlingau, which his father's domestic servant Marie Baldreich had ‘aryanised’, was restituted to him in 1948; it was no longer possible to return the liquidated business. In 1950, Hugo Engel dissolved his Parisian art dealership and acquired a trade licence for dealing in oil paintings at Reisnerstraße 12 in Vienna’s 3rd district. This licence remained valid until 1956. His claim under the Federal Restitution Act (BRÜG) filed in the FRG in 1959 for the loss of 170 paintings from the shop on Zirkusgasse was rejected, as the works of art exported by Faltis had travelled directly from Vienna to Sweden and therefore did not meet the requirements of the BRÜG. Herbert Engel worked as an art dealer in England and became a British citizen. In 1956, he changed his surname to Terry-Engell.

In 2013, six objects that Engel had acquired from Richard Neumann in 1941 were restituted from the holdings of the Musées nationaux récupération (MNR) to Neumann's heirs.