All EU legislation shall continue to apply until such time as the UK actually leaves the EU. After that date, much will depend on what deal if any is agreed with the EU.
If EU competition law ceases to apply after the UK leaves the EU, there will be limited change, as agreements, cartels, abuses or mergers which restrict competition and have an effect on trade within the EU will continue to be subject to EU competition law by virtue of the 'effects doctrine'. Domestically, the Competition Act 1998, which prohibits anticompetitive agreements and abuse of a dominant position, and the Enterprise Act 2002 which applies to cartels and mergers, applies irrespective of EU law.
If EU state aid rules cease to apply, then subsidies by the UK Government to industry would be more widely permitted. Less strict WTO rules would apply.
As regards public procurement, if a post-Brexit deal does not offer access to the Single Market, UK contractors will no longer benefit from equal treatment and transparency when it comes to the award of contracts by public authorities or utilities in the other 27 member states. Similarly, however, UK authorities and utilities will not be under such obligations to contractors from the other 27 member states when it comes to the award of UK contracts.