In short: probably not.
For companies looking to protect their brands in the UK and across Europe more generally, it was, until recently, most cost effective to register trademarks with the European Union Intellectual Property Office (EUIPO). Upon registration, the relevant intellectual property would be protected throughout the EU, including the UK.
Once Brexit was announced, it was agreed that any EUIPO trademarks granted by 29 March 2019 would automatically receive protection in the UK post-Brexit (via a corresponding UK trademark). However, this protection would not extend to trademarks that had not fully progressed through the registration process by 29 March 2019.
As this deadline approaches, there is a greater risk that new EUIPO trademark applications will not complete in time to guarantee UK protection post-Brexit. As a result, companies looking to protect their brand in the UK and throughout the EU are strongly advised to file for both UK and EUIPO trademarks moving forward.
It may be that a more favourable deal is reached before Brexit takes place, but given the overall uncertainty, this should not be relied upon.