We are unable to issue you two consecutive J visas.  If you want to participate in two J-1 programs without leaving the United States in between, you should book an appointment for a visa for the first J-1 program. If you qualify for the first J-1, you can choose to carry both DS-2019 forms for your two programs and two SEVIS fee receipts when you travel to the U.S. Port of Entry.

You can discuss both programs with the CBP officials at the Port of Entry and they may choose to take both into account when considering your authorized period of admission. Please note that we can offer no assurances in advance on whether this option will be successful.  A second option you may consider in this scenario is to wait until you return to the UK from your first program. You can then make a second J-1 application at the embassy.