Why the does 'tu' get pronounced 'tyu' in British English?

Despite being a native Brit, I've always found it an oddity that words like "tutor", "tube", "tumour", and "duty" are pronounced as "tyutor", "tyube", "tyumour", and "duty" in British English. For me, this doesn't seem phonologically simpler/easier at all, and given that it seems to be absent in American English (possibly other forms of English too?) for these sorts of words (beginning with "tu" or "du"), I would suppose it's a somewhat recent innovation in historical linguistic terems, almost surely after the Middle English period. So that's my first question: when and where did this pronunciation originate, and as a bonus, why?

Now, there are certain words that have this "y" sound inserted even in American and other forms of English, though I can't think of any where it's inserted in the "tu" or "du" sound. e.g. "beautiful", "cute", "futile". Notably, this pronunciation seems much more natural to me in these cases, unlike in the "tu" and "du" cases. Of course, the pronunciation of these words in the original Latin (or even the intermediary French when applicable) suggests that this innovation happened within English, and most likely within England itself. (N.B. I can't think of any Anglo-Saxon words with this mode of pronunciation right now, but quite possibly there are some too.) So, my second question is: is there some clear linguistic/phonological reason why the insertion of the 'y' sound is more natural in the non-"tu"/"du" cases, and did this phenomenon originate earlier in the English language?

Wikipedia has what seems to me to be a very complete write-up on the pronunciation of words like "tutor," "news," "brew," that are not pronounced with a "y" sound for many speakers, even though the spelling and history would suggest that pronunciation. This phenomenon is named "yod-dropping," as "yod" is one name for the "y" sound. In general, it is possible to predict which words are affected, although the rules are fairly complex and there are some areas where there is variation even among speakers of the same regional variety. For this reason, I won't try to list them all here and risk giving a simplified, wrong picture–you can find them over at Wikipedia, or if you look at phonologists' work on describing "yod-dropping."

So to answer your first question: the pronunciation with "y" is the original one, and has the same origins as it does in words like "feud" or "fume" where both British and American English speakers standardly have a "yoo" sound. The pronunciation without "y" is newer, but I've had a hard time finding actual dates given that correspond to the timeline of this change. It appears that yod-dropping occurred earlier or later depending on the particular phonetic environment, and the change is still in-progress.

Regarding the second part of your question: the consonant "y" is pronounced towards the front of the mouth. For many American-English speakers, there is a tendency for the ordinary "oo" sound to be pronounced more towards the front of the mouth after coronal consonants, a class that includes /t, d, n, s, z, l/ among others (Source: The Atlas of North American English, Sound Changes in Progress, the fronting of /uw/ after coronals). It seems possible to me that this tendency first led to confusion between "yoo" and "oo" after these sounds (for example, in the pair of words "do" and "dew"), and then led to what phonologists call neutralization: a complete lack of contrast.