I'm slightly confused by the questions, so will post more than is probably necessary.

If you purchase a MileSAAver Business ticket it will apply to AA Business, IB Business Plus, BA Club World, etc. As others have discussed in many threads, MileSAAver awards are frequently available on BA because you pay horrendous YQ charges you will not pay on other AA partners (there are some on IB, but not as high).

Now about what I think refers to MileSAAver and upgrades? You can not upgrade an award ticket.

If you upgrade a purchased fare, on AA it upgrades from Coach (including MCE, as it's still Coach) to Business; if you upgrade on BA, you upgrade from World Traveller (coach) to World Traveller Plus (premium economy) because that IS the next and separate class of service (beats the slats out of MCE, though). You would need a WT+ ticket to upgrade to Club World on BA - but again, you can not upgrade an award ticket, and to upgrade on BA with AA miles you need to purchase in the highest fare bucket available for that class - often more than a discounted fare in the next class of service anyway.

BA has no MCE equivalent - MCE are coach seats with extra seat pitch; World Traveller Plus is a different product, wider seats with greater pitch, etc. See here.