Land Registry records are in the public domain. Anyone can see who owns your house and who has a charge registered against the property (such as a mortgage lender) simply by paying £3 for a copy of the title register here:
https://www.gov.uk/search-property-information-land-registry
(The title register should indicate whether the property is freehold or leasehold).
As Woofgang says, it's definitely worth checking whether the mortgage company have lifted their charge against the property. (If they haven't you won't be able to sell it). When I paid my mortgage off the building society managed to lose my final payment (of around £35k) and then tried to repossess my house because I'd not been making monthly repayments! I eventually thought that I'd got it all sorted out but then, over a decade later, when I tried to sell part of my garden to a neighbour I found that I couldn't because the building society had forgotten to remove the charge against the property from the title register!