The binary alloy of titanium-tungsten (TiW) is an established diffusion barrier in high-power semiconductor devices, owing to its ability to suppress the diffusion of copper from the metallisation scheme into the surrounding silicon substructure. However, little is known about the response of TiW to high temperature events or its behaviour when exposed to air. Here, a combined soft and hard X-ray photoelectron spectroscopy (XPS) characterisation approach is used to study the influence of post-deposition annealing and titanium concentration on the oxidation behaviour of a 300~nm-thick TiW film. The combination of both XPS techniques allows for the assessment of the chemical state and elemental composition across the surface and bulk of the TiW layer. The findings show that in response to high-temperature annealing, titanium segregates out of the mixed metal system and upwardly migrates, accumulating at the TiW/air interface. Titanium shows remarkably rapid diffusion under relatively short annealing timescales and the extent of titanium surface enrichment is increased through longer annealing periods or by increasing the precursor titanium concentration. Surface titanium enrichment enhances the extent of oxidation both at the surface and in the bulk of the alloy due to the strong gettering ability of titanium. Quantification of the soft X-ray photoelectron spectra highlights the formation of three tungsten oxidation environments, attributed to WO$_2$, WO$_3$ and a WO$_3$ oxide coordinated with a titanium environment. This combinatorial characterisation approach provides valuable insights into the thermal and oxidation stability of TiW alloys from two depth perspectives, aiding the development of future device technologies.