No. The Post Office is not a private company, it's a public limited company with the government as sole shareholder.
It was changed from a government department to a statutory corporation in 1969. It was then changed to a public limited company in 2000.
Furthermore:
- Post Office Ltd owns and runs Post Office Counters Ltd which runs the post office branches. This is the company that uses Horizon (since 1999)
- Royal Mail delivers mail to addresses, and owns the Postcode Address File. Royal Mail was separated from the Post Office and privatised in 2013. It has never used Horizon.
Horizon is an EFTPOS/accounting system, nothing to do with mail delivery. It was introduced to the Post Office in 1999 after Fujitsu/ICL were originally commissioned by government to build an accounting system for the Benefits Agency, and it was so awful and buggy the Benefits Agency rejected it, so the government asked them to retool it for the Post Office.