The voice seems to be saying "wŷr" (w = consonant, y = vowel), but the spelling seems to indicate w = vowel, y = consonant (or glide/semivowel).
Are both pronunciations found?
No. The two have different pronunciations.
This may be clear if you go to www.gweiadur.com (register for free if you haven't already) and look up gŵydd and gwŷdd. You can click on the blue speaker symbol next to each word to hear a natural voice recording of their pronunciations.
-ŵy- sounds more like /oy/ as in boy but from nearer the front of the mouth, like a quick /oo-i/, whereas -wŷ- sounds more like /wee/.
So I'm just hearing the computer voice wrong (or its pronunciation is a bit off with this word).
Can native Welsh speakers tell the phonetic difference between 'oer' and 'wyr' (my phone can't include the diacritic needed but you know what I mean)? Or is it the same as English 'rain' and 'reign' where the original differences have changed as phonetics have shifted over the years and became the same sound, even though spelling shows a difference?
They are different sounds - you don't need to be a fluent speaker to hear the difference.
oe- is further back in the mouth than ŵy-. The sound of oe- is about the same as the 'oi' in 'loin' in British English (although in some dialects it will tend more towards a British 'or' sound)
ŵy- here is more of a British English 'ooee' sound - but with the 'oo' and the 'ee' being very short and run together. It is more at the front of the mouth than oe-.
If you go to www.gweiadur.com (register for free) there are natural voice recordings of many words. Try listing to oer, oen, poen, lloeren and ŵyr, gŵyl, gŵydd, pwyll. Look up each word and then press the small blue speaker icon.
oe- is further back in the mouth than ŵy-...It is more at the front of the mouth than oe-.
It's better to think of the tongue as being higher for wy than oe rather than being further back.
To say Welsh oe and English "oi", you start with the lip position for English "o" in "for" and then the mouth closes somewhat and the tongue moves forward. To say Welsh wy, the tongue start higher up in the position for English "u" in "put". The lips are more rounded and closer together than for the start of oe. Then the tongue moved forward (like it did for oe) and the lips spread. It can be imitated like you say by saying long English "oooo" then follow it straight by "eeee" - wy is essentially what happens when you say these sounds together in quick succession.
I like your suggestion to listen to native speakers on Gweiadur. There are some examples in this video too.
I agree with everyone that is saying the pronunciation is off. But what I noticed is that the normal speed pronunciation is considerably different than the slow speed...and both hit my ear wrong.