Most of the airspace issues are going to be either between Syria and the UAE (France), Syria and Cyprus (UK) and Syria and wherever the aircraft carrier/fleet is (USA).
Erdoğan is very protective of Turkish Airspace (considering the shooting down of the MiG that took a short cut over Turkish territory), and although Turkey has a NATO base at Incirlik with the USA, I can't see Russia or Syria annoying Erdoğan any more than they need to. That said, the Allies, as NATO members, can use Incirlik, though this would probably be for refuelling or emergency diversion, rather than the departure base for an offensive mission, which would be a political hot potato for Turkey.
Apart from the rhetoric and the doom-sayers in the media, providing Russian forces aren't directly attacked deliberately, it probably won't escalate dramatically (excuse the partial alliteration!), though you may see a few more high altitude contrails from transiting aircraft, quite apart from the regular civil traffic.
I don't think you'll need your tin hats and gas masks - save them for when you're at home!