I was told that if a Muslim is for example in prison, tied to a pillar, unable to make wudhu or tayammum, his / her awrah is not fully covered for salah, doesn't know the Qibla, suppose also has najasah on his / her clothes and / or body and unable to remove that najasah BUT knows it is time for a particular prayer then he / she MUST pray (making the movements as best as he can). Is this correct?
Allah (ﷻ) said, إن الصلاة كانت على المؤمنين كتابا موقوتا (Indeed, prayer has been decreed upon the believers a decree of specified times) - Qur'an 4:103. Just from being mentioned in the Qur'an specifically, you know it is an important condition.
Furthermore, the Prophet (ﷺ) said, أفضل الأعمال – أو العمل - الصلاة لوقتها (The best of' the deeds or deed is the (observance of) prayer at its proper time) - Sahih Muslim. There's various other hadith recorded in An Nasai, Tirmidhi, etc. which say that this particularly is beloved to Allah (ﷻ).
Thus, you will find it being said, (eg, by IslamQA) that time is the most important condition.
However, everything you described above is an exception and likewise there is some leniency and forgiveness from Allah (ﷻ) when it relates to time as well. eg, in some circumstances the Prophet (ﷺ)/Sahabah had to delay praying after the fixed time due to war, as one example.
Another situation in your prisoner example may be that the prisoner is locked in a prison where there are no clocks or windows. But not knowing time doesn't prevent him from estimating times and praying. So it just depends on how you spin the scenario.
But as a general rule, whatever you are able to do MUST be done.