My first thought here was that Marie Carter of the Independent is either rather obtuse or willfully ignorant in regards to comparing the mass-euthanizing of British pets to a continent-wide genocide of 6 million people, both in the headline as well as in the article. It's a pretty shitty style to draw a moral equivalence between euthanizing pets in WWII and, you know, the actual Holocaust that happened in the same time frame. But anyways...

The actual Holocaust against people didn't start outright but was preceded by years of legal and social discrimination of Jews in Germany and the territory it acquired. I have written before on the subject rather extensively here and the gist of it is that the genocide the Nazis committed was the radicalization of previous years of escalation of rhetoric and discriminatory measures taken against a group that had to socially, politically, and legally defined in the first place and had to be made reviled in the public eye.

To that end the 1930s saw a flurry of legal discriminatory measure, the best known probably the Nuremberg Laws and the laws that lead to Jews in the state bureaucracy losing their jobs. But these were far from the only measures. Jews lost their citizenship in 1935 and were relegated to "wards of the Reich" (Schutzbefohlene des Reiches); in 1936 they were banned from certain public parks, restaurants, and swimming pools; in 1938 from cinemas, theaters, and other public entertainment venues and they were also barred from owning electrical/optical equipment, bicycles, and typewriters; in 1939 their radios were confiscated and a special curfew for them was established; they were banned from using most of the public transport and from shopping except for certain hours; and so on.

As I wrote in the linked answer by 1941 "genocide was in the air" and with swaths of the USSR conquered and the systematic murder of Soviet Jews having begun, measures also began for the deportation of German Jews. This included most notably a prohibition of Jewish emigration from Germany altogether but also a bundle of measures in preparation of their planned deportation to the Eastern territories.

Among these measures was a Reich decree in September 1941 that forbid Jews from owning cats, dogs, and birds. Not only were they forbidden from acquiring them but Jews who owned pets were now legally forced to either give them away or bring them to a vet to kill them. This was in effect both in the Reich as well as in the General Government where Jews who had been forced into Ghettos had already been forbidden from bringing their pets and could only take with them a very small amount of their possessions. What they couldn't take with them was sold by the Nazi administration though it is unclear what happened to pets in this instance. Most likely they were either given directly to their neighbors or simply freed or killed.

Further than that no coherent policy is known since in most occupied countries, it was the collaborationist administration that handled Jewish property of the victims of deportation. So f.ex. in the Netherlands, France, and Greece it is unclear how the local administration handled pets though given that most of these countries were forced to introduce provisions similar to Germany, it is very likely that they instituted the same policy and Jews were forced to deal with their pets prior to ghettoization or deportation.

In the USSR and Serbia on the other hand where Jews were killed wholesale by the Einsatzgruppen and the Wehrmacht in actions where whole villages were rounded up, there are reports of pets being either directly shot by Wehrmacht and the Einsatzgruppen or simply set free to roam the country side.

So, in summary, where deportations were planned and happening, Jews were forced to deal with their pets prior to deportation and ghettoization while where large scale shooting actions occurred, the pets were either killed or simply set free.