We have stayed at all three and they each have pro's and cons.
Bathrooms
Biggest single issue may be bathrooms. AKL-Kidani and BLT 2 bedrooms each have 3 bathrooms. Huge plus. WL only has two.
Transportation
WL has boats to the MK and busses to everything else. BLT has the walkability factor to the MK, as well as direct Monorail access which allows you to also get to EPCOT. Busses to HS and AK. AKL is busses to everything,
but the busses are dedicated to AKL only. Not true at BLT or WL.
Now, some folks complain about how far AKL is from everything--personally, it has never bothered me (seriously, we are talk an extra 5 minutes in the bus each way, at most).
Decor/Atmosphere
BLT is very unique, but it is very modern. If that does not appeal to you, then it will probably be a big turn-off. WL itself has really captured the grand western lodge feel. It is a beautiful resort that feels isolated while only being a few minutes from the MK Personally, you can't beat AKL. Some folks complain that the rooms are to dark, I have never found that to be the case (frankly, all of the green features at the resorts make them all a little on the dark side). We like the solid/substantial feel of the villa furniture, the subtle infusion of Lion King into the decor and the ANIMALS! You simply cannot go wrong with a savanna view. Again, some complain about the long hallways; given how much we walk every day, that has never phased me.
Restaurants
All have great choices, but again, I think I'd have to give the nod to AKL. Not only do you have one of the most popular buffets (Boma's), you have one of the most under appreciated (though it's following is growing) restaurants in Sanaa's and one of the premier restaurants on Property (Jiko's). Jiko's may have supplanted WL's Artist Point as our go-to date night restaurant. That said, Artists Point is also outstanding. WL's other restaurant, Whispering Canyon, is fine, but nothing that excites me that much. With BLT, you have an outstanding restaurant in the California Grill (assuming it will be open when you get there--currently closed and undergoing major renovations). Chef Micke's is standard buffet fair, very loud and very popular. It does have character meals if that is important to you. You also have the Wave--can't comment, never eaten there.
Pools
All three are pretty darn good and all three have zero entry (I am pretty sure). At AKL, you have two excellent pools to choose from, so you can mix things up. You can also do that at BLT, but I find the Contemporary pool to be plain boring.
Misc.
CM's at AKL are beyond top notch. It is often hard for me to reconcile the fact that it's CM's work for the same company as some of the other CM's I have interacted with over the years. Can't sing high enough praise. Add to that, AKL is the only place outside of EPCOT, where you have international CM's specifically placed in their home country/continent. Makes a big difference.
So, you really can't go wrong with any of them. We have stayed at all three and would again. That said, when we go back this Christmas,
we are doing AKL.
Good luck with your choice. Keep us posted