Is my acceptance into the hospital unlawful (haram) and an injustice to my colleagues who are more senior than me, especially since the hospital had previously rejected them due to full capacity, then accepted me through nepotism (wasta)?

As long as the hospital administration accepted the request without deception or fraud, there is no blame upon you in that, nor is there any injustice to others. These jobs are matters of fortune and provision, and you followed the correct procedures to obtain it, and you were successful and accepted. So, use what Allah has provided you with in His obedience, and be grateful to Him for what He has bestowed upon you.