Yes, you can file a writ petition in the High Court under Article 226 of the Constitution for expeditious disposal, as the right to a speedy trial is a fundamental right under Article 21. For the Senior Citizens Act proceedings, the High Court can direct the Maintenance Tribunal to decide your mother's eviction application within a strict timeframe, as the Act is a beneficial legislation meant for urgent relief. For the criminal case, since it has been pending for five years, you can file a petition under Section 528 of the BNSS (formerly Section 482 CrPC) seeking a direction to the trial court to conclude the trial expeditiously. The Supreme Court has held that prolonged and unexplained delay violates the right to a speedy trial. You can also seek a direction to the trial court to conduct the trial on a day-to-day basis. Given your mother's health conditions, the High Court is likely to prioritize the matter. You should file a single writ petition seeking directions for both the Senior Citizens Act proceedings and the criminal case to be expedited, supported by medical documents and the pending status of the cases.