Research institute reluctant to provide recommendation letters. Should I persist asking them?
Professors are very busy. I wouldn't take "no reply" as "don't want to do it", your email may well be sitting in a folder labeled "look at these later" somewhere...

Send both professors another email, and begin the subject line with "Time sensitive" to get their attention (eg. "Time sensitive: Letter of recommendation"). You might also try calling, although if all you have is an office phone number, many profs are even worse with their office voicemail than they are with emails.

When you email, include your CV, and a separate document with some bullet points about yourself and your achievements (both during your time with them and since your graduation if applicable). The less work they have to do to write the letter for you, the more likely they'll be able to fit it into their busy schedule.

Keep trying -- but do NOT worry too much if you can't get a letter, that does NOT automatically disqualify you. Unlike with a PhD advisor, who only works with a few students and works with them for several years (it would be weird not to get a reference there), a masters' thesis advisor works with many more students and for a shorter period of time, so it's not that unusual to not be able to get a reference (especially if it's been a few years - mine barely remembered me when I contacted him for a reference three years later).