Hello aspirant,
With a JEE Main rank around 225,000, you are well-positioned to secure admission to Mechanical Engineering in reputable government-aided and private colleges in Lucknow under AKTU (formerly UPTU). For example, IET Lucknow, a top government college, typically fills its Mechanical seats up to around 68,000–75,000 in early rounds and extends to about 75,000 to 90,000 in later rounds. Since you're beyond that range, IET may not be feasible.

However, University of Lucknow (LU) via UPTAC often opens Mechanical at ranks up to 225,000 to 270,000 in the final round, and your rank should comfortably get you in there. Private AKTU-affiliated colleges in Lucknow like LIT, SRMCEM, BIET, RRGI, GITM, etc., tend to close Mechanical seats at ranks between 600,000 and 1.4 million, making them viable with your rank.

So your best options are Mechanical Engineering at University of Lucknow through state counselling, or picking core branches in mid-tier private AKTU-aligned colleges.