Concurrent writes block each other, but following one-time "pragma journal=wal", read and writes can work concurrently, and you can expect x2-x5 performance (or ~2% degradation, depending on use pattern, but most people see x2 performance).
The cost is that access through network e.g. NFS or SMB in wal mode is impossible (but you shouldn't have done that anyway), and that you can't just ship the sqlite file - ship a dump/backup instead, or you'll have to do recovery on the wal file you sent.
Of course, it's a good idea to use pgsql from the get-go; but SQLite deserves more credit than it gets, and is much more capable than it is usually assumed to be.