What is the value of a champagne bucket marked wm rogers 851?

Not much, unless you have one of the few patterns made of real sterling.

A silver-plated 1883 FB Rogers Co coffee set with both tea and coffee pots, tray, creamer and sugar bowl may sell for as much as $200-300, depending on the style, condition and desirability, but a coffee pot alone is worth much less. Typical FB Rogers tea pots sell for $10-20 online, although there are a few rare or unusual patterns (like a certain footed pot from 1960) that are difficult to find and may bring $40-$100 on resale, even without the missing pieces.

Bear in mind that most 1883 FB Rogers Co silver is actually silver plate over metal, not solid sterling. The products are good quality, however, due to being quadruple-plated, which makes them more durable and better able to hold their finish than some other companies' products.

Having said that, be aware that Replacements, Ltd., indicates there were a few patterns made of real sterling. The registered pattern numbers are 118, 142, 144, 149, 164, 250, 400, 700 and 1636. The genuine silver version will have the Word "Sterling" stamped on the bottom of each piece. If it doesn't say "sterling," it isn't. Naturally, sterling will command a higher price than silver plate; for example, several thousand dollars for a tea set.

Also be aware that 1883 (and the crown) is part of the maker's mark, indicating the year the company was founded, not the year the teapot was made. You can estimate the actual age by checking the model name or number stamped on the underside of the pot.