Obviously we can't be entirely sure what will happen there but yes there is the potential there for an earthquake which could generate a tsunami tho from rough scratchings it would need to be 2000 times more powerful than the highest anticipated to generate the smallest of what would be classed a tsunami - makes me glad we live in the UK right now.
I'm no expert but I believe that tho the mid-atlantic ridge is splitting away rather than converging that the geological makeup and motion is not uniform and there is the potential for events around the Azores whether quake or volcanic to generate a tsunami.
EDIT: Looks like I'm a bit wrong - in 1931 it did cause a "tsunami" tho not what we'd really class as one today.
Generally large earthquakes occur either along the boundary of subduction zones (convergent, one plate going beneath another, like this recent one) or strike slip faults (San Andreas, where plates slide past each other). Other areas generally don't have the forces needed to create bigger quakes/fault movements, which is why 6.5 is BIG in our region of the world (stress is concentrated at plate boundaries and dissipates the further you go away from them, the ones in the north sea (Dogger bank) are reactivated weak zones from earlier periods of history*). You don't really get big quakes at transform faults either due to the location they are in (divergent settings), as you said they occur because sections of the ridge are spreading at different rates**
The main thing to remember about tsunamis is that it's not the magnitude of the earthquake that is the issue, it's the displacement of the fault (or sea floor) either side of it. Uplift of a few metres along several hundred/thousand km will cause a large tsunami.
The biggest threat to the UK is probably a tsunami caused by land slide, lickily we're probably pretty safe for that as well.
Yes saw a documentary on that a few years ago - Mega-Tsunami. Cannot remember exactly what Island it is, but if the fault slips, they believe virtually half the island will collapse into the sea causing a Mega-Tsunami which would race across the Atlantic hitting the East Coast of the USA. If that happened most of the major US Cities, such as New York, Boston etc would be wiped out!
Apparently all rubbish, essentially made up by one guy that wanted to make a name for himself and latched onto by the media. In scientific circles it's essentally seen as a crack pot idea... Good documentary though, I remember seeing it as well (was a Horizon one).
* Specifically, most during the Jurassic, when the North Sea was trying to become an ocean (rifting), it failed, obviously, but it did give rise to all oil and gas riches in the North Sea.
** Not necessarily different rates as such, just at different times.