Can I extend my one-year UK master's degree to two or three years due to visa issues, and if not, how can I stay in the UK longer after completing it?

Your course length is set by your university's academic structure, not by immigration rules, so you cannot lengthen a genuine one year master's simply to gain more time in the UK; the routes to staying longer all sit outside the course itself.

Legitimate Ways to Stay Longer

• The Graduate Visa: your main post study option, currently 2 years if you apply on or before 31 December 2026, or 18 months from 1 January 2027, with 3 years for PhD or doctoral graduates; it allows work in most jobs without needing sponsorship and cannot itself be extended.
• Switching to a Skilled Worker visa: if you secure an eligible sponsored job offer, you can switch from Student or Graduate status, since the rules allow switching where you have completed your sponsored course or your job starts after the course finishes.
• Progressing from master's to PhD: a genuine academic progression, not a way to artificially extend your stay, that can lead to a further Student visa period and a subsequent three year Graduate visa on completion.

When You Might Genuinely Need a Student Visa Extension

If your concern is that your Student visa expires before your university allows you to finish your dissertation or course, that is a different situation from simply wanting extra time. Speak with your university's international student or visa team before your current permission expires, since the Student visa rules specifically provide mechanisms for extending permission where a course legitimately runs longer than planned.

My Advice

Use your master's year itself to build toward a Skilled Worker outcome rather than waiting until after graduation to start job hunting, since the shortened 18 month Graduate Visa from 2027 leaves less runway than the earlier 2 year period. Start networking and applying for sponsored roles several months before your course ends.