There is no bad season as such for an attic conversion. Interior work continues year round regardless of the weather, so you are not tied to summer to get a good result. What is weather dependent is the external side of a dormer or hip-to-gable, where the roof is opened up.
For those external stages, timing matters more. Scaffolding, roof-open work and anything that exposes the structure are planned around the forecast, so the roof is not left open in heavy rain. In practice this means the external stages are scheduled sensibly rather than avoided in any particular month, and Irish weather can be changeable in any season, so the forecast matters more than the calendar. A Velux conversion is less exposed to this, since the roof is not opened up in the same way, which makes it more flexible to schedule.
Winter is perfectly workable. Once the structure is closed in and made weathertight, insulating, first fix, plasterboarding, skimming, second fix and decorating all happen inside and are unaffected by the cold or the rain. Many people are glad to have the work done over winter so a new room is ready for the year ahead.
The more useful thing to plan around is lead time, not season. Design, structural engineering and, where it applies, planning permission all take time before the build starts, and planning in particular can add several weeks. If you have a target date in mind, the sooner you book a free assessment, the more freedom we have to schedule the weather-sensitive stages well.