The more you know about your collection, the better the price you will get. If you are going to approach dealers or auction houses, it is best to provide a description up front in terms of the makers and style of paperweights you have. Otherwise they may not even respond. Since you are in the UK, my advice is to get a list of dealers from the Paperweight Collectors Circle and start with them.
I am in the US and I have purchased collections that large, but it is a hard thing to deal with, for both the seller and the buyer. You might start with your best weights and try to get a reasonable price for them. Then donate the remainder to a charity or consign them to a local auction, knowing there will be little return. I don't recommend selling them on eBay unless that is something you have always wanted to try. It will take you years to sell them all.