According to Mina's LGPE concept art, she is between the ages of 13 and 15 in those games. We also know from a scene in Moon that Trial Captains must abdicate their positions when they turn 20, so Mina, being an active-duty Trial Captain, can be no older than 19 in the Alola games. (While we're at it, since we also know that you must complete the island challenge to become a Captain, and that you can embark on the challenge once you turn 11, that gives us a firm 11-19 ballpark for all active Trial Captain characters. I would imagine that Mallow and Kiawe are on the older end of that, as they both maintain other professions alongside their Captainship duties, while Sophocles is clearly younger, as I would say are Acerola and Lana. Ilima is probably somewhere in between.)
Considering how Mina is rendered in LGPE, being as tall as Chase/Elaine, and that the original Kanto protagonists are all taller than you and her, the implication I think is clear is that they are meant to be a little older than you, especially as they have all already gone on their respective journeys. And Red, at least, we know to have been 11 when he became a Pokémon Trainer.
The tricky part when it comes down to pinpointing the age of Red/Blue/Green in LGPE and the age of Mina in SM is figuring out where, relative to the conventional Pokémon timeline, LGPE fits in. We have always known that the Kanto and Hoenn narratives occur in close proximity, and then 3 years pass (by which point Red has turned 14) in time before the Johto and Sinnoh narratives. After that is a dubious gap until the events of Black & White, which are in turn followed by two consecutive 2-year passages between the events of B2W2/XY and the events of SM or USUM. I do not think it is necessary to squeeze LGPE into the 3-year gap between the original Kanto events and the original Johto events, since everything clearly unfolded a bit differently in the LGPE timeline. Red and Blue already went on a journey, but the Pokédex has only just now been invented as of Chase/Elaine's beginning, and they are the ones who go on to bust Team Rocket, which implies that Red and Blue had no significant prior dealings with the Rockets.
I think the simplest way to square this is to say that Red becomes a Pokémon Trainer in the same calendar year regardless of which timeline we're looking at - let's say 1996 as a pure hypothetical. In one timeline, he comes up against Team Rocket. And 3 years after that, in 1999, he battles Ethan at Mt. Silver. But in another timeline's 1996, he and Team Rocket never cross paths for whatever cosmic variable reason. This is the LGPE timeline, and in it, Team Rocket will only become more active later on down the line, well after Red has defeated the Indigo League and battled with Ethan, if Ethan existed. Ergo, Red will be older than 14 when that time comes. Similarly, Blue will take up control of the Viridian Gym at an older age than he did in the original timeline, because the impetus for him doing so simply didn't even occur until later in this timeline.
So the events of LGPE, in relation to the overall Pokémon timeline, probably take place closer to those of Black & White than to those of the Johto and Sinnoh games. Now, I tend to assume that the gap between Johto and BW is about 5 years (based on the manner of the foreign Rocket Grunt's child, although I do sometimes entertain the idea of longer stretches). But sticking with a 5-year estimation, and knowing that Mina can be no older than 19 in the Alola games, we can deduce that a 19-year-old Mina in SM would be 15 at the time of BW (as there are four cumulative years between that story and SM), about 10 at the time of Johto/Sinnoh, and 7 at the time of Kanto/Hoenn.
Meanwhile, Red, who was 11 during the Kanto story and 14 during the Johto story, would be approximately 19 during the first Unova story, 21 during the Kalos story and the second Unova story, and finally 23 during the Alola story. So now we can establish a cap - the only points at which Mina could be between 13-15 without being over 19 in SM would be as follows:
Kanto/Hoenn (1996) = Red 11, Mina 5/6/7
Interim Year 1 (1997) = Red 12, Mina 6/7/8
Interim Year 2 (1998) = Red 13, Mina 7/8/9
Johto/Sinnoh (1999) = Red 14, Mina 8/9/10
Interim Year 1 (2000) = Red 15, Mina 9/10/11
Interim Year 2 (2001) = Red 16, Mina 10/11/12
Interim Year 3 (2002) = Red 17, Mina 11/12/13
Interim Year 4 (2003) = Red 18, Mina 12/13/14
Unova 1 (2004) = Red 19, Mina 13/14/15
Interim Year 1 (2005) = Red 20, Mina 14/15/16
Unova 2/Kalos (2006) = Red 21, Mina 15/16/17
Interim Year 1 (2007) = Red 22, Mina 16/17/18
Alola (2008) = Red 23, Mina 17/18/19
Based on that timeline, the earliest point at which LGPE can take place would be two years prior to (2004), the calendar year of BW. Any earlier, and she would not be at least 13 as her concept art indicates. And without injecting more years into the Johto/Unova gap (which you could do, but for the sake of simplicity, I will not), the latest point at which it could take place would be in (2006), the calendar year of B2W2/XY. Any later than that, and she would be over 19 by the time of the Alola story and therefore no longer eligible to serve as a Captain.
Now, some particulars. We don't know when exactly Mina became a Trial Captain. It is possible (albeit less likely in my opinion) that she already is one by the time of LGPE. As I established in my first paragraph, you only need to be at least 11 in order to take on the island challenge, and once you have completed it, you can be appointed to serve as a Captain. However, I think it would be odd for her to take leave of her official post in order to go on a casual trip to Kanto. Sure - it is not completely out of the question, but still, it inclines me to think that she becomes a Captain after her stay in Kanto.
That being said, since she is already older than 11 in LGPE, I find it probable that she has already started (and maybe even completed) her island challenge, as she is clearly a skilled Pokémon Trainer even at that point. I would just assume that there is a gap in between her finishing her trial and her appointment as a Captain.
Something else potentially worth noting is that Poni Island, where Mina's home and eventual trial site are located, was without a kahuna for a few years, because the previous kahuna, Hapu's grandfather, died unexpectedly. We know from USUM that Nanu took up those responsibilities until Hapu was chosen by Tapu Fini to assume the role, but this does cause me to wonder if Mina was appointed by Hapu's grandfather and has been serving since then, or if she was chosen by Nanu acting in lieu of an actual Poni Island kahuna. A detail that I would point out is that in SM, Mina has not even decided what she wants her trial to be like yet. This suggests to me that she is a relatively new appointee, and also that she is very much not on the cusp of retirement, which makes her being 19 at that time seem a lot less likely to me. Therefore, the 17-18 range strikes me as being a lot more favorable for her age in SM. Looking at the timeline above, that really only leaves us with two options - either LGPE takes place during BW's calendar year (2004), or in the year before that (2003). Without inserting more years into the Johto/Unova gap, those are the only times are which she can match the age from her LGPE concept artwork while also being younger than 19 in SM. Of the two, I find it a little more comfortable to say that LGPE takes place in the year before the events of Black & White, because 19 feels just a touch too old to me for the Red and Blue that we meet in LGPE. It doesn't make a huge difference, but them being 18 feels more a little more fitting to me for some reason.
