The official advice from the DVLA is as follows: 'From 20 May 2018, all vehicles other than public service vehicles, constructed or first registered more than 40 years ago which have not been substantially changed within the last 30 years will be exempt from MoT testing. These changes only apply to vehicles in Great Britain, the changes will not apply to vehicles in Northern Ireland.' So yes, it will be legal to drive your car after 20 May, even though it doesn't currently have an MoT certificate. However, your car will need to be roadworthy - otherwise, your insurance will be invalidated.