We present a study of the environments of 17 Lyman-α emitting galaxies (LAEs) in the reionisation-era (5.8 < z < 8) identified by JWST/NIRSpec as part of the JWST Advanced Deep Extragalactic Survey (JADES). Unless situated in sufficiently (re)ionised regions, Lyman-α emission from these galaxies would be strongly absorbed by neutral gas in the intergalactic medium (IGM). We conservatively estimate sizes of the ionised regions required to reconcile the relatively low Lyman-α velocity offsets (Δv_Lyα < 300 km s⁻¹) with moderately high Lyman-α escape fractions (f_{esc, Lyα} > 5%) observed in our sample of LAEs, suggesting the presence of ionised hydrogen along the line of sight towards at least eight out of 17 LAEs. We find minimum physical ‘bubble’ sizes of the order of R_ion ∼ 0.1–1 pMpc are required in a patchy reionisation scenario where ionised bubbles containing the LAEs are embedded in a fully neutral IGM. Around half of the LAEs in our sample are found to coincide with large-scale galaxy overdensities seen in FRESCO at z ∼ 5.8–5.9 and z ∼ 7.3, suggesting Lyman-α transmission is strongly enhanced in such overdense regions, and underlining the importance of LAEs as tracers of the first large-scale ionised bubbles. Considering only spectroscopically confirmed galaxies, we find our sample of UV-faint LAEs (M_UV ≳ −20 mag) and their direct neighbours are generally not able to produce the required ionised regions based on the Lyman-α transmission properties, suggesting lower-luminosity sources likely play an important role in carving out these bubbles. These observations demonstrate the combined power of JWST multi-object and slitless spectroscopy in acquiring a unique view of the early Universe during cosmic reionisation via the most distant LAEs.