The Wikipedia article you linked to states that L. Victor Allis showed in $1994$ that black wins on a $15\times15$ board. The "Gomoku Terminator" site you link to has an image of the upper portion of a board with $15$ columns. Thus it seems that this program merely does what was known to be possible in $1994$, and this has no bearing on the open question of the $19\times19$ board. Allis' thesis states that Gomoku used to be played on $19\times19$ boards because that's the size of Go boards, but that the $15\times15$ board has now become the standard.
You should clarify whether you wish to differentiate between positions based on en passant, castling, and whose side it is to move. Francis Labelle and others have used the term "chess position" to indicate a board state including the above information, and "chess diagram" to indicate a board state not including the above information, i.e. just what pieces are on the board and where. In neither case is information for drawing rules like the 50-move rule or the triple repetition rule included.
The best upper bound found for the number of chess positions is 7728772977965919677164873487685453137329736522, or about $7.7 * 10^{45}$, based on a complicated program by John Tromp; according to him, better documentation is required in order for the program to be considered verifiable. He also has a much simpler program that gives an upper bound of $4.5 * 10^{46}$.
For chess diagrams, Tromps simpler program gives an upper bound of about $2.2 * 10^{46}$; he does not say what bound is obtained by the complicated program, but it is probably a little less than half (since side to move doubles the bound, whereas castling and en passant add relatively little), so likely about $3.8 * 10^{45}$. More information at Tromp's website
The best bound published in a journal was obtained by Shirish Chinchalkar in "An Upper Bound for the Number of Reachable Positions". I do not have access to this paper, but according to Tromp it is about $10^{46.25}$. Although it refers to "positions", it could very easily be a bound for the number of diagrams.
As for lower bounds, they are much more difficult, since a given position could be illegal for very subtle reasons. Wikipedia has claimed that the number of positions is "between $10^{43}$ and $10^{47}$", but I think it is unlikely that the lower bound has been proven.
I would guess that the actual number of diagrams is between $10^{44}$ and $10^{45}$, but this is purely speculation.
Best Answer
The classic reference on this is the paper "Computing a Perfect Strategy for nxn Chess Requires Time Exponential in n" by Aviezri Fraenkel and David Lichtenstein; unfortunately, I can't find the (30-year-old!) paper online anywhere and while I think I have the proceedings it came from, it's buried in a pile of boxes. Of course, your board isn't finite, but your initial piece configuration is; this means that if there's any computable bound on the needed board size to resolve an initial position, then chess isn't Turing-complete. (If the status of all positions of diameter $d$ could be determined using boards of size no more than $f(d)$, then they can be determined in $\mathrm{EXP}(f(d))$ time by the EXPTIME completeness result, and thus wouldn't be Turing-complete). I vaguely recall seeing something about such a result, but unfortunately some quick searches aren't turning anything up. The expert on the mathematics of chess is Noam Elkies, so hopefully he can spot this and chime in...