I think a point needs to be made about the original question in the thread;
Weather avoidance "in THE TMA." Not at altitude. That is a different set of circumstances and I think the regs already cover most contingencies in thas regard.
At altitude our TAS is very high, therefore it behoves all of us to ensure timely weather avoidance. Transport aeroplanes will come apart inside these Cb walls at 8 miles a minute! So do whatever you have to do, get off track, climb, descend away from traffic, broadcast intentions, etc, etc. (All things that have been rightly noted in this thread.)
However, in the TMA, I think a little more mental latitude is called for. Our speed is lower, our airspace is extremely limited, and we are in close proximity to both other aircraft, all trying to avoid, and of course, terrain.
ATC have a job to do, so try and make it easier for them... give them options."I require heading xx for xxmiles to avoid" That helps him plan. If he can't help you he may give you an alternative (then again, he may not, so be ready for it) If all fails, you may have to be prepared to ride it out, or hold clear. With only minor exceptions, worldwide, I've yet to see weather below the freezing level that will bend a transport aeroplane. Sure it'll shake you up a bit, so batten down the crew early if you know you're going to be in the thick of it and be prepared to ride it out at minimum penetration speed.
If it is really that bad, and you dn't want to push on, then hold clear of the weather (at least ATC can plan around you as a fixed point in space then) The Final approach requires more caution of course, as you are configured with a lower g-tolerance and windshear is your biggest threat, so by all means AVOID./ Hold/whatever.
But for goodness sake, don't make a bad situation worse by self vectoring in proximity to other traffic...who may be 'self vectoring' unannounced themselves! ATC will hardly be able to rescue you then: Imagine a multiple TCAS RA event going on WITH weather avoidance by other traffic! As to the A7700 option.... I'm not convinced that'd be a good idea. Certainly NOT in most 2nd and 3rd world ATC environs.( I stand to be corrected, happily if it becomes ICAO promulgated practice, of course) It'd cause total confusion and almost certainly would not get you the result you need. Remember this is a TMA situation here... you simply don't have the space.