The only significance of 1 July is that is the date that the UK has decided to bring into force the Annexes to the Aircrew Regulation (e.g. Part-FCL, Part-MED, etc.). The Regulation itself will become EU law on 8 April and there is nothing that anyone can now do to prevent that. The option to delay the implementation of the Annexes was agreed by the EC EASA Committee last year but depends on an amendment to the Aircrew Regulation being in place by 8 April - this is by no means certain and if the amendment isn't in place, the Annexes will also become law on 8 April. This is one of the reasons why there is confusion (not to say a certain degree of panic) within the CAA.
From the date that Part-MED becomes EU law (either 8 Apr or 1 Jul depending on whether the amendment is issued in time) medical examinations will be conducted in accordance with Part-MED. However, any medical certificates issued in accordance with JAR-FCL prior to the implementation date of he Annexes (whatever it is) will be deemed to have been issued in accordance with Part-MED. There is nothing anywhere that suggests that the holder of a valid medical certificate issue under JAR-FCL 3 should have to do anything other than renew it in accordance with the existing arrangements.