When I (jokingly) affect a non-American accent , I pronounce these like "bowda" and "showda"
What do you mean by a "non-American" accent? There are dozens, probably hundreds, of identifiable non-American accents, and not all of them say "bowda".
You might be interested to know that you have actually nailed the London* accent! I'll explain. In the traditional London accent, "oh" sounds, as in "boulder", tend to get wider, so they get closer to the vowel sound in "how now brown cow". Closer, but not all the way. The next bit is the 'l', which tends toward a tight "ooo" or elongated "w" sound.
Finally, the last vowel. In most of England we would pronounce "boulder" with a "duh" at the end. In London, it tends towards a "dah" sound.
So, in most of England, boulder is pronounced bow
1-l
2-duh
3.
1: "bow" to rhyme with bread dough
2: a dark (but not silent) 'l'
3: a short and fairly light "uh" sound
In London, it's more like:
1: "bow" gets wider - let's say half-way between "cow" and "dough"
2: the 'l' becomes "ooo"
3: a short "ah" sound
*I refer to a London accent, but sadly two-thirds of the population of London are not white British, which means the traditional London accent is in retreat. Also, there is the old Cockney accent. It sounds to most people like an exaggerated version of the traditional London accent, although I'm sure it's more complicated than that.