bk-tree | BK-tree Java library | Dataset library

BK-trees
Kudos to Ben Hoyt and his link to the issue which I will draw from. That being said, the first observation from the mentioned issue is that the BK tree isn't exactly logarithmic. From what you told us your usual d is ~6, which is 3/10 of your string length. Unfortunately, that means that if we look at the tables from the issue you will get the complexity of somewhere between O(N^0.8) to O(N). In the optimistic case of the exponent being 0.8(it will likely be slightly worse) you get an improvement factor of ~100 on your 10B entries. So if you have a reasonably fast implementation of BK-trees it can still be worth it to use them or use them as a basis for a further optimization.

The downside of this is that even if you use 1000 trees in parallel, you will only get the improvement from the parallelization as the perfomance of the trees depends on the d rather than on the amount of the nodes within the tree. However even if you run all the 1000 trees at once with a massive machine, we are at the ~10M nodes/tree which you reported as slow. Still, computation wise, this seems doable.

A brute force approach
If you don't mind paying a little I would look into something like Google cloud big query if that doesn't clash with some kind of data confidentiality. They will brute force the solution for you - for a fee. The current rate is $5/TB of a query. Your dataset is ~10B rows * 20chars. Taking one byte per char, one query would take 200GB so ~1$ per query if you went the lazy way.
However, since the charge is per byte of a data in a column and not per complexity of a question, you could improve on this by storing your strings as bits - 2bits per a letter, this would save you 75% of the expenses.
Improving further, you can write your query in such a way that it will ask for a dozen strings at once. You might need to be a bit careful to use a batch of similar strings for the purpose of the query to avoid clogging of the result with too many one-offs though.

Brute forcing of the BK-trees
Since if you go with the route above, you will have to pay depending on the volume, the ~100-fold decrease in the computations needed becomes ~100-fold decrease in price which might be useful, especially if you have a lot of queries to run.
However you would need to figure out a way to store this tree in a several layers of databases to query recursively as the Bigquery pricing depends on the volume of the data in the queried table.
Building a smart batch engine for recursive processing of the queries to minimize the costs could be fun optimization excercise.

A choice of language
One more thing. While I think that Python is a good language for fast prototyping, analysis and thinking about code in general you are past that stage. You are currently looking for a way to do a specific, well defined and well thought operation as fast as possible. Python is not a great language for this as this example shows. While I used all the tricks I could think of in Python, the Java and C solutions were still several times faster. (Not to mention the rust one that beat us all - but he beat us by algorithm as well so it's hard to compare.) So if you go from python to a faster language, you might gain another factor or ten or maybe even more of a performance gain. This could be another fun optimization exercise.
Note: I am being rather conservative with the estimate as the fuzzywuzzy already offers to use a C library in the background so I'm not too sure about how much of the work still depends on the python. My experience in similar cases is that the performance gain can be factor of 100 from pure python(or worse, pure R) to a compiled language.