Yes
My answers refer to "immediately" as in the sense of being able to go straight to the gym upon arrival in that specific town, without having to complete a certain task.
Kanto (Red/ Blue)
Pewter City & Brock
Cerulean City & Misty
Celadon City & Erika
Fuchsia City & Koga
Source
Kanto (HG/SS)
Vermillion City & Lt. Surge
Saffron City & Sabrina
Celadon City & Erika
Fuchsia City & Janine
Seafoam Island & Blaine
Source
Kanto (FR/LG)
Pewter City & Brock
Cerulean City & Misty
*Vermillion City & Lt Surge
Celadon City & Erika
Fuchsia City & Koga
Source
Kanto (Gold & Silver)
Vermillion City & Lt. Surge
Saffron City & Sabrina
Celadon City & Erika
Fuchsia City & Janine
Pewter City & Brock
Seafoam Islands & Blaine
Source
Johto (Gold & Silver)
Violet City & Falkner
Azalea Town & Bugsy
Goldenrod City & Whitney
Ecruteak City & Morty
*Cianwood City & Chuck
Mahogany Town & Pryce
Blackthorn City & Clair
Source
Johto (HG/SS)
None. All require a certain task in order to be accessible.
Source
Hoenn ( Ruby/ Sapphire/ Emerald)
Rustboro City & Roxanne
Dewford Town & Brawly
Mauville City & Wattson
Lavaridge City & Flannery
Petalburg City & Norman
Fortree City & Winona
Mossdeep City & Twins Liza & Tate
Source
Sinnoh (Diamond/ Pearl/ Platinum)
Eterna City & Gardenia
Canalave City & Byron
Snowpoint City & Candice
I had to base this region off memory so, if I've missed one or added one that shouldn't be here let me know below please. My doesn't state that you have to go to the Oreburgh Mines in order to fight Roark, so I knew I couldn't go off of it completely.
Source
Unova (B/W & B2/W2)
Striaton City & Trio Evaluators
Nacrene City & Lenora
Castelia City & Burgh
Nimbasa City & Elesa
Icirrus City & Brycen
Aspertia City & Cheren
Virbank City & Roxie
Castelia City & Burgh
Nimbasa City & Elesa
Driftveil City & Clay
Mistralton City & Skyla
Opelucid City & Drayden
Humilau City & Marlon
Source
Kalos (X/Y)
Santalune City & Viola
Courmaline City & Ramos (Forgot he was a Leader)
Laverre City & Valerie
Anistar City & Olympia
Source
*Means it required you to have an HM, in terms of it's first usage; onwards I found it unnecessary to mark others.
Hopefully I've Helped!