"Applied Mathematica: Getting Started, Getting It Done" by Shaw and Tigg is 20 years old now, but it has an information handling orientation and I learned a great deal from that.

Almost any problem in Mathematica can be done at least a dozen different ways, at least a couple of which are completely incomprehensible.

You could turn the BOM information into a tree and walk the tree to count the total items used. You could use pattern matching and repeatedly replace items to create the count. You could use searching or selection to repeatedly extract items from a table. There must be other completely different ways of doing this. I thought about using a vector inner product to calculate the result but didn't spend the time thinking about that to be able to finish it up. It would be possible to create a queue, load the queue with the top level demand items and then repeatedly pull the next item from the queue, if it has children entries in the BOM table insert all those into the queue, otherwise put it into an output pool, until the queue is empty, then perform a sort of Union on the output pool to merge all the quantities for each unique childID. All entries would need a scalar multiplier to indicate the number needed for that particular step in the process. This was actually the idea behind my early attempt at a solution and only part way through did I switch to using := to support the calculations.

This is a good example of an information or data processing problem that involves some minor calculation.  I'm new to Mathematica and have been trying to evaluate how useful it is for information processing as opposed to mathematical calculations.  There are many references to data handling or data manipulation on the Wolfram site but there are few examples of it so it is hard to tell what are real capabilities and what is marketing hype.  This problem makes a good test case for sorting that out.  Is the posted solution  the only way to handle this type of problem in Mathematica?  For data processing problems (that do not involve curated data from Wolfram) are the only options to use the Database Link or to treat the data as Mathematica identifiers?

Ooo.. Bad programmer.

The smallest modification I can make is to change the last line toIs that sufficient? I am not at all certain that has not broken something else.

Test this carefully before you depend on it. Please let me know if there are any other errors you find.