Ah yes, the effects of age, I could tell you a lot about that. But I’ll try to keep focused on in-game effects.
“I’ll regret this later”
Penalties to abilities that simply make the character weaker (like those found in older versions) are frustrating for a player, and they are bland, and don't really evoke the feel of old age. These rules simply never provided any fun, or much depth. It’s no wonder they didn't come forward to 5e.
Effects that limit the endurance a character has, that come into effect after some initial exertion, evoke the effects of age more poignantly, while letting the character still “relive past glories” in brief but glorious combat.
So very tired
The rules for exhaustion in the DMG can be utilized to simulate the fatigue of advanced age, and they come pre-play-balanced for you. A venerable character (depending on age, at the discretion of the DM) may gain a level of exhaustion from a single combat, and require a short or long rest to recover from it.
I just can’t seem to rest like I used to
Speaking of rest, an older character needs more. Simplest thing would be to double the amount of time needed to attain the benefits of the rest, but it doesn't add much color to the game, if the players are simply obliged to say “OK, then we rest for 16 hours.”
I find it’s better to “nerf” the effects of the rest, granting back fewer hit dice, hit points - or even fewer spell slots, if mental ability has been affected. That allows the oldster(s) to “try to keep up” while providing a nagging reminder they are really too old for this sort of thing.
(If you ever played 4E, this was like a non-heroic NPC taking a rest. Unlike the PC’s, the NPC would not wake up in the morning fully healed of all wounds.)
Another nice surprise
I would also suggest that these or any effects of old age should come as a surprise (especially for the prematurely aged). They always do.
My (game) experience with old age effects
I’ve used these rules for adjunct NPCs - I haven’t prematurely aged a PC. The players thought they were fair and interesting rules. It accentuated the power of the (young) PC’s, without making the NPC useless. The oldster slept/rested while the characters did ancillary stuff. It might be different for a PC. I suspect a player won't be overjoyed with having to “take it easy” but it's similar to other “curses.”
Since initially answering this, I spoke with one of my players about the age rules we used. She commented the rules made her feel protective of the NPC.
It's up to the DM...But the DM has no Guide.
The effects of aging, from slowing down all the way up through death, are not currently listed in the 5th edition of D&D. It appears that the designers either forgot to include them, or, slightly more likely, that they left the issue up to each table to decide.
The Aging is Real
That said, the aging from the ghost's horrifying visage ability is real rather than cosmetic, because it specifies that the effect ages the creature, not just alters its appearance to look older. High level Monks (Player's Handbook p79) are immune to the aging portion of the effect, as are high level Oath of the Ancients Paladins (who don't appear to die of old age, by omission) (p87).
Best Answer
Unclear
Wild Shape says your statistics are changed to match that of the animal you are transforming into, but 'age' is not a statistic listed anywhere in a creature's statblock, so at that point, you're out of RAW options and have to decide what RAI you use.
You can go three ways with this:
If you run with option 1, I argue that the animal form is the creature that gets aged. If you run with option 2, I argue that both forms would get aged. If you go with option 3, I argue that the druid form is the one that gets aged.
Ultimately, it doesn't matter all that much because aging effects are few and far between. I've personally always used option 3, because it allows fun stuff like crippled druids who turn into animals to get working arms and legs back, but whatever you decide should be fine.