It used to be pronounced vuh-GARE-i, but it's changed. As @Tim above stated:
(GenAm) IPA: formerly /ˌvəˈɡɛɹi/, now commonly /ˈveɪˌɡəɹi/
It's usually pronounced VAY -guh-ri, but there are still people who pronounce it the "other way", as can be seen here
The /ˈɛryən/ pronunciation is just a result of English phonology processing a foreign borrowing that starts with the letters AR.
Aryan is a borrowed word in all languages outside the Indo-Iranian subfamily of Indo-European. The rest of the world pronounces it as some variant of [arjan], which comes, as noted, from Sanskrit ārya /a:ryə/ 'compatriot'. Therefore, /'aryən/ is a perfectly acceptable English pronunciation, and the only acceptable one when using the term in its modern Indian sense.
Any use of Aryan (outside scare quotes) that refers to Germany or white racism is a result of romantic interpretations of 19th century German linguistic scholarship (e.g, Grimm's Law), which unearthed the prehistory of the "Indo-Germanic" (as I-E was then called, from names of its Eastern- and Westernmost families) languages. It was all very exciting, apparently. See also Wagner, Mad King Ludwig, Neuschwanstein, German Empire.
The AHD of IER says that Skt ārya comes from the PIE root *aryo- 'Self-designation of the Indo-Iranians'; other descendants of the same root are Iran and, surprisingly, Eire -- Celtic languages sometimes retain PIE roots that are otherwise lost in the Centum group.
None of these are English words, and so English treats them the same way it treats all borrowed words -- it changes the pronunciation until it tastes right. That's all.
Edit:
I almost forgot, another reason to pronounce Aryan /'aryən/ is because Arian /'ɛriən/ usually refers to Arianism, a very important variety of Christianity that was the religion of the Ostrogoths in Italy, and the Visigoths in Spain. The only Gothic texts known to exist are translations of various parts of the (Arian Christian) New Testament.
Best Answer
The way in which people pronounce words depends on regional accent. Natalie, the presenter in these clips has a Northern Irish accent, hence the 'graind'. When you're learning a language, you are usually presented with a 'polite' form of it. What is known in England as 'received pronunciation', and in this version of English, the ou in 'ground' would rhyme with 'wow!'. The 'Thish'would be pronounced without the final 'h'. People who have very fluent English as a second language often struggle with regional accents which vary widely.
There is also the question of the politics of language, in which some ways of speaking are deemed more acceptable than others. This is still a live issue. There was a recent controversy in which a school teacher was told by a schools inspector that she had 'better lose that northern accent'.
I am learning conversational Greek, and I was told by a Greek friend that we were being taught 'very posh Greek'.