The way most solicitors work is that they will have hundreds of clients, and that is not an exaggeration, and for things that don't have a court deadline, like they are not trying to get a brief bundle together for a court date next week. Or have an entry in their calendar for them to look at a file. Everything gets done pretty much in the order of the stack of paper on their desk.
So you are essentially in a queue, even reading an email. It's not so much as "money in account" that automatically trigger a review of the file. More like they will review each file every week, once a week and then do what can be progressed on that file. This way, every client gets equal attention, every file gets reviewed.
When they get a letter, you can imagine they can get like 30, 50 letters every morning. Depends on the size of the firm, the large ones the mail room will opens all the letters, collate them into department or even solicitors and then distribute them in a stack per fee earner.
Now days the mail room might even scan each letter into the system, upload digitally into a file and it's for the fee earner to flag that job when the letter is actioned.
I would expect an average 7 day reply time scale, when everyone is at work...