Essentially, there's LOTS of interest in both London and Birmingham guns. The keys are grade and condition. Even a gun from a relatively unknown gunmaker from Birmingham can command serious money, if it is a high grade in excellent condition.
The broken specimens aren't worth much because you can't shoot them.
The rule of thumb is to judge each piece on it's own merits, and avoid generalizations about any particular brand (brand in this context takes into account that makers names become trade names, trade names can be bought and sold, and materials, quality and workmanship depend on grade and date of manufacture).