British politicians pronouncing "hear, hear" oddly
Solution 1:
The speaker featured at the referenced moment is Kirsten Oswald, a member of the Scottish National Party. Seated near her are other Scottish MPs such as Alex Salmond, the former leader of the Scottish National Party. It is this grouping of Scottish MPs who are calling out "hear, hear" in support of Ms Oswald, and it is the Scottish accent that gives rise to the unusual sounding pronunciation being noted in your question.
An English accent will pronounce hear, hear in a manner more familiar to American ears.
(The debate appear to be taking place in a committee room located in the Palace of Westminster.)
Solution 2:
I think you are probably hearing a version with a semivowel /j/ followed by word-final schwa, which Geoff Lindsey says is a current variant of the "NEAR" vowel in British English. This would not be a speech impediment.
Lindsey's blog post The demise of ɪə as in NEAR (April 21, 2012) says
In the earlier standard/reference accent of British English, Received Pronunciation, words like NEAR contained a centring diphthong, ɪə. This was a vowel which glided from the lax quality ɪ to the quality ə within a single syllable. [...] Although British dictionaries still use “/ɪə/” in their transcriptions, a lax diphthong of this type is now rather old-fashioned.
Contemporary NEAR
In contemporary Standard Southern British (SSB), we hear tend to hear either
a long pure vowel, the monophthong ɪː; or
a form in which the tense FLEECE vowel is followed by schwa, which we could write as ɪjə or, with traditional symbols, as /iːə/; this form can plausibly be considered to comprise two syllables.
Many speakers use both forms. For such speakers, NEAR is what John Wells has termed varisyllabic, and the long monophthong in (1) can be thought of as derived from the disyllable in (2) by ‘smoothing’.
The blog post has a number of audio examples you can listen to.
Edit: I think the audio in the Youtube video linked to in the question does sound a bit like the sound file in Lindsey's blog post, although maybe some of the speakers also have final consonantal /r/. I can't exactly tell. The woman who is speaking most of the time in that section of the video certainly does have consonantal /r/ in this environment, since she seems to have a Scottish accent, but I don't know if all the other people saying "hear, hear" have the same accent she does.
A Scottish accent of course is not "southern British," but I think the / ɪjə/ pronunciation Lindsey mentions may exist to some extent outside of the South as well (or /ɪjər/ with a final consonantal /r/).
Solution 3:
This "hear, hear" is a stock phrase with a very long history (back to the 1700s) of use in the British and Commonwealth parliaments (it is also commonly used in the Australian parliament to this day, where a slightly odd pronunciation is heard here as well). It is short for "hear him, hear him", and apparently became common because to clap/applaud was not permitted during debate in the houses of parliament.
- https://en.wikipedia.org/wiki/Hear,_hear
- http://www.oed.com/view/Entry/85048#eid1883652
I would argue that, because the long tradition of this phrase, and its very specialised area of use, the pronunciation has drifted a bit from perhaps the common pronunciation out in the broader society.
What is perhaps interesting is that the phrase isn't used in the US. I guess it just happened that the people involved in the early US Congress hadn't been exposed to the UK practices, even though it looks as though the use of 'hear, hear' was already established at that time in the UK.
Solution 4:
It's just how they pronounce the words (actually 'hear, hear') when using them as an exclamation of approval during a noisy debate in which a lot of privileged people having dined and drunk well get to clap each other on the back for still being able to string together one or more vaguely coherent sentences. I would imagine this dates back to the beginning of Parliament itself.
I'm British, I've peered at them through the t.v. lens a few times.
The rest of us rarely , if ever, use this form of exclamation with such pronunciation for fear of being mistaken for one of them.