Using a PoD somewhen between 1154 and 1214, how plausible is it that the medival
Angevin Empire could survive to the present day (still ruled by the Plantagenet line)?
How wide might it extend, in the 21st century (assume that, at
minimum, the Empire contains Great Britain and the western half of France what is France in OTL)?
What would the world be like?
Those last two questions are virtually impossible answer. The simple reply is "it could go any way". If you set up conditions to favour the Angevins, they could take the French throne reasonably quickly, then if they so wished, spend several centuries interfering with Germany, and easily take the Imperial throne, and/or large chunks of Germany. Perhaps all, over time. After that, with such a vast empire then talks of recreating the Charlemagnian Empire would arise and Europe might not be powerful enough to prevent Italy and Poland falling. After that, the theory of a united Christendom might well arise, in one way or another. This megalith country could be a brutally expansive colonial power. The sky would be the limit over time.
On the other hand, it could end up in frequent conflicts with France, as IRL. Even if the Empire doesn't fall as RL, it could face paralysis from within from infighting and never gain an upper hand on the French. Under that condition, it would likely be put onto the defensive in a way that it could never recover from. The empire could be reduced to England and strips of coastline in Normandy and Gascony.
And really it could go anywhere in between. The bigger an empire gets, the more likely to collapse spectacularly, and there's all manner of ways it could fall apart and be reduced to a level seen only a few centuries before. If it doesn't have a massive setback like that, then at any point it could stagnate like the Spanish Empire, and spend the rest of its existence gaining and losing tiny territories which mean very little, unable to expand but too strong to fall. In addition, sudden expansions are usually responded to by antagonism from alliances of neighbouring states which could put an effective stop on expansion and even reverse recent acquisitions.
In short, with such an open-ended story, a POD set 800 years ago and an end-date of now, just about anything could happen.