How could Jews integrate into Western modern society back when Saturday was a normal working day?

I found a relevant passage in this (German language) booklet on women within Bundism.

Note that his concerns the Rayon within Zarist Russia, I don't know if this is western enough for you.

... war im ausgehenden 19. Jahrhundert ein jüdisches Proletariat entstanden, welches im Wesentlichen aus jiddischsprachigen HandwerkerInnen bestand. In diesem Zeitraum machte die jüdische Arbeiterschaft circa 15% der jüdischen Gesamtbevölkerung aus. Die überwiegende Mehrheit der jüdischen ArbeiterInnen war in Kleinbetrieben bzw. erweiterten Handwerksstätten tätig, die in jüdischem Besitz waren, und in denen (fast) ausschließlich JüdInnen angestellt waren. Gerade aufgrund dieses Arbeitsumfelds war eine Assimilation in die Mehrheitsbevölkerung nahezu ausgeschlossen

my translation:

[the conditions outlined above] gave rise to a jewish proletariat in the ending 19th century, which mostly consisted of jiddish speaking tradespeople. In this timeframe, workers made up about 15% of the jewish population. The large majority of jewish workers was employed in small shops and extended artisan manufactureers, which were in jewish possession and where (almost) exclusivley jews worked. Especially because of this working environment, assimilation into the majority population was almost impossible

On the other hand, David Graeber notes in an essay on a totally different topic:

Even in the days of Karl Marx or Charles Dickens, working-class neighbourhoods housed far more maids, bootblacks, dustmen, cooks, nurses, cabbies, schoolteachers, prostitutes and costermongers than employees in coal mines, textile mills or iron foundries.

So it appears integration into mainstream society was possible near the precarian bottom rungs or in the educated upper middle class, but not into the respectable working class.

(Edit: Whoops, sorry, I assumed you were USian, but I see you're UKian, so this is more of an aside than an answer.)

According to Wikipedia, the five day work week was adopted across the US in the 1940s, and accommodation for Jewish workers actually contributed to its evolution (emphasis mine):

In 1908, the first five-day workweek in the United States was instituted by a New England cotton mill so that Jewish workers would not have to work on the Sabbath from sundown Friday to sundown Saturday. In 1926, Henry Ford began shutting down his automotive factories for all of Saturday and Sunday. In 1929, the Amalgamated Clothing Workers of America Union was the first union to demand a five-day workweek and receive it. After that, the rest of the United States slowly followed, but it was not until 1940, when a provision of the 1938 Fair Labor Standards Act mandating a maximum 40-hour workweek went into effect, that the two-day weekend was adopted nationwide.

Reform Judaism does not view the Sabbath as irrevocable holy day and the movement which started mid 19th century grew considerable so that your remark "they were indistinguishable from non-jews European people and led similar lives outside of religious matters" is only valid for this branch. I would even say that the integration and non-conspiciousness of Jews in the modern world (they still follow many rituals and traditions, so you can recognize a Reform Jew if you are a good observer) is only possible with the creation of Reform Judaism. Saturday is therefore not a problem for integration.

As reaction the ultra-orthodox Jews (Haredim) were moving in the opposite direction which means separation from other denominations, denial of the modern world and strict observance of Jewish law. The Haredim did not integrate into the western world at all and their distinctive appearance means they are very easy to recognize. The relationship between Haredi and Reform Judaism are for this reason a bit...strained.

The rest of the Jews who were more conservative, but would not decline to participate in the modern world are the Modern Orthodox branch. They try to balance the observance to traditional laws and the requirements of modern life. According to the 1990 National Jewish Population Survey a majority of 78% Orthodox jews consider themselves Sabbath observant.