We present the results of extensive computation of the Thermal Pulse phase AGB evolution of stars of metallicities in mass fraction 2 x 10(-4) less than or equal to Z less than or equal to 0.01, for those masses in the range 2.5 less than or equal to M/M-. less than or equal to 6, which suffer the Hot Bottom Burning (HBB) phase. The evolution is fully computed, by assuming a mass loss rate consistent with the observations of the Magellanic Clouds lithium-rich stars, and modelling convection with the Full Spectrum of Turbulence model by Canuto and Mazzitelli. The results are discussed in the framework of their importance for the evolution of proto-Globular Clusters, whose spectra show that the stars are very probably formed from matter contaminated by the ejecta of these stars, or have accreted it after formation. The main results we find are the following: 1) for metallicities Z less than or equal to 10(-3), masses above similar to4 M-. suffer complete CNO cycling in HBB, so that they show at the surface the result of this process, and the oxygen abundance is reduced; 2) most models suffer the third dredge up. Although carbon is processed to nitrogen by HBB, the oxygen burning is so strong in the lowest metallicities (2 x 10(-4)) that carbon becomes more abundant than oxygen: in other words, low-metallicity intermediate mass stars may show up as carbon stars due to the drastic oxygen burning; 3) if Globular Cluster stars are contaminated by matter processed through these phases, we must expect a non negligible helium enhancement in their composition: from a Big Bang abundance Y = 0.24, e.g., we might expect an abundance Y = 0.28. This may have no practical consequences if pollution concerns only the external parts of the stars, but is very important if the stars formed as a whole from a helium rich environment. 4) The lithium yields, although not important for galactic chemical evolution, are very interestingly close to the initial Big Bang abundance: processing by HBB is the only way in which we can obtain substantial amounts of gas which have gone through full CNO burning, but preserve a reasonable abundance of lithium.