How do I write a mysql query to combine data from related tables into a single summary?
Yes, there is a way. Just add an identifier column to the selection list. E.g.:

select count(warr_id), year(prod_date) AS prod_year, month(prod_date) AS prod_month, 'W' as type
from product_t, warr_t
where prod_date is not null and serial_number = product_look
group by year(prod_date), month(prod_date)

union

select count(serial_number), year(prod_date) AS prod_year, month(prod_date) AS prod_month, 'P' as type
from product_t
where prod_date is not null
group by year(prod_date), month(prod_date)


When you build the chart for the query use the type column for different series