Have I understood this right?
Pretty much, yes.
Assuming that no agreement is reached and the UK leaves the EU on 29 March 2019, it immediately becomes a 'third country' in EU law. Consequently, the UK CAA will not be able to issue any EU licences, ratings or certificates and all approvals issued under EU law by the UK cease to be valid. Training completed up to midnight on 29 March will remain valid for the issue of EU licences, ratings and certificates but any incomplete training will have to be completed in an EU member state and the relevant licence, rating or certificate issued by such a state. Existing licences issued by the UK CAA prior to 29 March will remain valid in all respects as UK national licences or, if the holders medical records have been transferred to an EU member state, as EU licences.
The worst affected will be those pilots wishing to fly for an operator established, residing or with a principal place of business in the EU. Generally speaking, a UK-licensed pilot flying a UK-registered aeroplane will not be seriously inconvenienced unless, of course, the operator of that aeroplane is based within the EU.
An ATO may be approved by only one EU member state. If, prior to 29 March, an ATO choses to seek approval from another member state, it is a matter of EU law that all existing approvals issued by the UK must be revoked. The same is true in the case of examiner certification, which must be transferred to the other EU member state. EU licences, ratings and certificates issued in the UK, other than examiner certificates, will continue to be valid until 29 March, at which point they cease to be recognised in EU law. If the medical records of ATO personnel are transferred to the approving EU member state prior to 29 March, the associated licences will continue to be recognised by the UK CAA for at least 2 years - a significantly more pragmatic attitude than that being shown by the EU.