Casino Royale is a reboot, confirmed both on the timeline and behind the scenes. It changes the relationships of characters and the timeline itself. The events of GoldenEye take place around 2010 now. The other movies would occur between Quantum of Solace and Skyfall also, I suppose. The codename theory is nonsense, so I'm discarding it as such. M being the same actress is a connection, but they have different names, Barbara Mawdsley in the GoldenEye script and Olivia Mansfield in Skyfall, so it's just a case of two alternate universe versions of M who look the same.
Let me prove that it's the same guy from 1962 to 2002.
Firstly, the dates in the films do place them definitely in their release years. Don't worry, that has a solution that doesn't involve super serums or anything, lol.
James Bond (George Lazenby) meets Blofeld for "the first time" after meeting him in the previous film (played by Sean Connery). This is because On Her Majesty's Secret Service was their first meeting in book canon. Bond was wearing a Japanese disguise in the previous film, so I guess Blofeld didn't recognise him (?).
In On Her Majesty's Secret Service (1969), James Bond marries Tracy in 1969, where she is killed in a drive-by shortly afterwards.
In For Your Eyes Only (1981), Bond (Roger Moore) takes flowers to Tracy's grave (George Lazenby's wife). The gravestone states that died in 1969. In the script and comic adaptation, it states that Bond and "Blofeld" last met 10 years ago, which matches with Diamonds are Forever (1971), starring Sean Connery. There's other connections between these three Bonds but this definitely places them in a single timeline.
In License to Kill (1989), it's mentioned that Bond (Timothy Dalton) married once, and it's a sore subject, tying in continuity from previous Bonds.
In The World Is Not Enough (1999), Bond (Pierce Brosnan) avoids the question of if he ever lost someone he loved, once again bringing Tracy up as a continuity nod. Further, Bond's family motto mentioned in On Her Majesty's Secret Service (1969), "the world is not enough", is referenced, although that was taken from Sir Thomas Bond, a real person.
In Die Another Day (2002), we get a scene with various gadgets and references to previous Bond films, suggesting that they are all part of the same history as that film.
Additionally, certain characters are recast even within a single Bond's series, or they stay the same actor even between Bonds. To say that a different Bond is a new continuity would essentially make most of Sean Connery's films different universes too, if we assume that a recast means a different reality. We know that's not the case though, so logically every film is a shared timeline until the explicit continuity reboot.
As for Bond's age, this is speculative, but if we apply Bond's age from the Moonraker novel (37) to the film timeline, we can apply that to every other film. It's not canon, but it's certainly interesting how well it plays out. The Bond films simply occur from his 20s all the way to his early 60s. It's not unbelievable in the realm of fiction. We can just apply this to the other characters in the series, like Monnypenny, Leiter, maybe Q and M, whatever seems the most reasonable compared to Bond's age.
1962 - Dr. No (Age 21)
1963 - From Russia with Love (Age 22)
1964 - Goldfinger (Age 23)
1965 - Thunderbolt (Age 24)
1966 - You Only Live Twice (Age 25)
1969 - On Her Majesty's Secret Service (Age 28)
1971 - Diamonds Are Forever (Age 30)
1973 - Live and Let Die (Age 32)
1974 - The Man with the Golden Gun (Age 33)
1977 - The Spy Who Loved Me (Age 36)
1978 - Moonraker (Age 37)
1980 - For Your Eyes Only (Age 39)
1982 - Octopussy (Prologue) (Age 41)
1983 - Octopussy (Age 42)
1985 - A View to a Kill (Age 44)
1987 - GoldenEye (Prologue) (Age 46)
1987 - The Living Daylights (Age 46)
1989 - Licence to Kill (Age 48)
1996 - GoldenEye (Age 55)
1997 - Tomorrow Never Dies (Age 56)
1999 - The World Is Not Enough (Age 58)
2001 - Die Another Day (Prologue) (Age 60)
2002 - Die Another Day (Age 61)