Playground Games applied a fix to the game which solves problem with glitched achievement. All you have to do is launch a game, select a car, wait few seconds and it should pop. Huge thanks to all people reporting this (especially NoHeroes94 who made extraordinary effort in the case) and Nitro Glitter who helped community with the fix!
In order to get the achievement for getting all 15 barn finds, you'll need to:
a) Play during all four seasons during Horizon Roster (so you need to complete initial tutorial with all four seasons depending on your influence)
b) Spend significant amount of money (10,000,000 CR) to buy Bamburgh Castle. You can find it on the east coast. This will give you barn find rumor.
Please note, there are two additional pieces of information worth mentioning. If you don't want to wait full 3 weeks, please scroll down to ROM3L's solution. It's great way of getting the achievement for impatient players. If you get all 15 and it doesn't pop, please scroll down to wookieepuppy's solution and read both it and its comments.
10 barn finds are 'regular'. You'll get them during your normal drifting, racing and doing stunts so don't focus on them. As mentioned above, another one can be earned after buying pretty expensive property (you'll want to buy this castle anyway as it will unlock another achievement for you:
As you enter Horizon Roster (you should be around level 20) you'll get your first seasonal barn find rumor. They can be distinguished by different colour of circumference and small icons showing what season they are from. Make sure you unlock them immediately. After the season has passed, you'll need to wait 3 more weeks until the whole process is repeated. For example, winter barn find can be found on small island in the middle of the lake so you won't get there until the lake is frozen.
Let's start with 10 regular barn finds. You can find them during any season so you don't need to pay extra attention to get them.
1) 1961 Jaguar E-Type S1:
2) 1993 Jaguar XJ220:
3) 1965 Mini Cooper S:
4) 1997 Lotus Elise GT1:
5) 1983 Audi Sport Quattro:
6) 1998 Subaru Impreza 22B STI:
7) 1962 Triumph Spitfire:
8) 1986 Ford Escort RS Turbo:
9) 1998 TVR Cerbera Speed 12:
10) 1966 MG MGB GT:
Below you can find locations of seasonal barn finds and property barn find:
11) Summer Season - 1962 Peel P50 (you'll need this for another achievement):
12) Autumn Season - 1960 Aston Martin DB4 GT Zagato:
13) Winter Season - 1973 Range Rover:
14) Spring Season - 1964 Ford GT 40 MKI:
15) Property - 1930 Bentley Blower: