First, the Ottoman Army always had a sizeable portion of Christian soldiers. I mean, always. If there is any point of a state like the Ottoman Empire waging wars against others for political and economic reasons, those are (beyond other, ideological reasons): getting taxes and recruiting soldiers. Now, you can recruit soldiers in three different ways. You can have a central, citizen-army like that of Napoleon, but that was not a thing until the 19th century. Alternatively, you can have a devshirme army. You take non-Muslim children from their families, teach them about Islam and make them learn Turkish (you achieve this through sending them to Turkish families with whom they grow up) and you have a janissary army. But again this is not what matters for us here, because these chaps become Muslim. The third option is what is relevant here; you conquer Serbia, you obviously find a quite complex social system with nobles and all that. What Ottomans did mostly is leaving the local nobles and the local system intact, welcoming but not directly encouraging conversion to Islam or Turkification. If there is a war, then you spread the word and expect your Serbian vassals (among others) to turn up, which they duly deliver mostly. In the Battle of Ankara in 1402, when many Turkish soldiers from the Anatolian principalities deserted the Ottoman Army, the Serbs stood ground and did not surrender until the Sultan himself (that would be poor I. Bayezid the Thunderbolt who was married to Olivera Despina Hatun, the sister of Serbian King Dejan Lazarevic) was captured by the Timur’s army. It would be an exaggeration to think these Serbs fought for a shared Ottoman identity (the marital connection and the pressure of a possible Hungarian invasion played a role, İnalcık argues), of course, but it is certainly not true that the Ottoman Army never had Christian soldiers. The same story applies to the conquest of Constantinople in 1453.
19th century was an important turning point. After the mid-19th century reforms dubbed tanzimat, you not only no longer have the janissaries (abolished in 1826), but you are now supposed to have a citizen-army including Muslims and non-Muslims alike. In practice, many non-Muslims found a way of circumventing this, by a re-invention of cizye tax, payment of which rendered a non-Muslim exempt from military duties. Yet, this did not apply to each and every member of the Ottoman society. So you have individuals like Kostaki Musurus Pasha, who served as an Ambassador in many places including London (quite famous there during the Russo-Ottoman War of 1877-78 as well!) and Athens (mind you, being an ethnic Greek himself). The translation bureau of the Ottoman Empire was largely made of Greeks until the Greek Independence and then of Armenians until the early 20th century, roughly speaking. Oddest of all for outsiders based on my experience, during the Balkan Wars in 1912, the Minister of Foreign Affairs of the Ottoman Empire was Gabriel Noradunkyan Efendi, an Ottoman Armenian (Findley, p. 239). These examples might easily be enumerated. What I wish to argue is that it is quite unthinkable that none of these individuals serving in the Ottoman Army, in various branches of Ottoman state quite to the best of their ability, did not feel in any sense as Ottomans. I strongly doubt that. This is not to deny that nationalism, separatism, and a mutual feeling of distrust among different ethnic and religious groups in general were on the rise. But the rise clearly did not include every member of non-Muslim communities.
References and Sources

Carter V. Findley, Osmanlı İmparatorluğu’nda Bürokratik Reform: Babıali, 1789-1922 (İstanbul: Tarih Vakfı, 2014). This one is also available in English, first published in 1980 as Bureaucratic Reform in the Ottoman Empire. It involves details on bureaucratic reforms of the 19th century and also provides more information on some non-Muslim members of the bureaucracy.

Halil İnalcık, ‘Bayezid I’, TDV İslam Ansiklopedisi, vol: 5, (İstanbul: 1992). Available online at: https://islamansiklopedisi.org.tr/bayezid-i