Radioactive waste glasses containing Mg have been shown to have much poorer (~one order of magnitude) aqueous durability in long-term, compared with those have a similar amount of Ca as the alkaline-earth constituent. This is the case between two non-radioactive research surrogates: MW glasses produced in the UK and SON68 glass produced in France. With the latter being the most widely researched and understood nuclear glass to date, advances in understanding and modelling MW glass dissolution require special account for Mg. Mg-containing clay precipitates observed on glass surface after reaching the residual rate regime have been thought to be the cause. However, precipitation initiated from early stages when the supposedly protective gel layer is forming and growing has not been studied. Solid-state NMR is extensively applied to characterise the altered glass surface in this thesis for its power in providing atomic-level structural insights with amorphous materials being amenable to it. Its quantitative nature also affords extra advantages over other spectroscopic techniques. Simplified MW25 glass (MgEM) and the series with various degree of Ca substitution were made to separately investigate the effects of Mg, firstly, on the pristine glass structure. The relative fraction of boron in 3- and 4-fold coordination changes as the relative amount of Mg (Ca) content changes in the glass. This is driven by the different structural roles (network forming/modifying) between the two alkaline-earth elements. The representative rare-earth element, La, also adopts different structural roles between MgEM and its counterpart CaEM, accordingly. However, the initial dissolution rates measuring their primary leaching kinetics do not reflect the varying structural features elucidated by $11$B, $27$Al, $23$Na, $25$Mg and $139$La NMR. Longer dissolution experiments up to 16 weeks are carried out to measure the extent of leaching with the feedback of secondary effects. The least durable MgEM glass is investigated in detail. Utilisation of $25$Mg-enriched MgEM glass as well as $17$O-enriched water as the leaching agent facilitates solid-state NMR experiments selectively characterising the alteration layers. Although suffering from low sensitivity, $1$H-$25$Mg and $1$H-$17$O cross-polarisation as well as $17$O multiple-quantum magic angle spinning NMR act as the surface probe. Coupled with $1$H and $29$Si NMR results, the altered glass and precipitates structures and the corresponding implications for glass dissolution kinetics are discussed at length. Aside from experimental results, NMR calculations based on ab initio density functional theory are performed to assist with explaining the complex $25$Mg and $17$O NMR spectra and hence identifying the precipitates.