The first, and only successful, private bank in Chertsey was Chertsey Bank, started in 1807 by Thomas La Coste with his brother George.  The bank survived until 1876 when it was taken over by Staines Bank.

£5 note issued by La Coste's Bank

Cheque issued by the bank after it had been taken over by Staines Bank - note the overprinting by Barclay's Bank, which had in turn just taken over Staines Bank

The other private bank to be formed in Chertsey was a short-lived affair.  Godley Bank was started by Henry Thomas Willats in about 1833.  The bank ceased trading when Henry died in 1835.

£5 note issued by Godley Bank (vignette showing King John signing the Magna Carta)