Records can be sold via eBay or other online sales sites. They may be suitable for reuse through a local reuse network such as Freecycle or Freegle. There are now a large number of reuse groups on Facebook.
If the actual records are broken, they can be put into your black/grey household waste bin and the sleeves can be put into your green kerbside bin. The sleeves can also be taken to your local Household Recycling Centre where they can be put into the paper bank.