LONDON—U.S. data center giant Equinix Inc. on Friday agreed to buy U.K.-based peer Telecity Group PLC in a cash and share offer worth £2.35 billion ($3.60 billion), squashing an earlier tie-up between Telecity and another European player, and underscoring the industry’s consolidation on the continent amid swelling demand for data and digital services.

Companies including technology and telecommunications firms are increasingly outsourcing data management and information technology handling to operators such as Equinix and...