Peter - when a joint registered owner dies the legal ownership, which we register, passes to the surviving registered owner. If any changes are required re that legal ownership then it would not be an Assent (AS1) you require as the executor of the deceased has no direct role to play in any such transfer of legal ownership.
IN the circumstances I would recommend seeking legal advice as to how the 'trust' has been set up in the way described and how best to now deal with the legal ownership and the beneficial interests of the 3 children. It may be that there is nothing more to do than apply to update the register re the death (form DJP plus certified copy of the death certificate) but if the best option is for the surviving joint owner to transfer the ownership to himself and the 3 children for example then it would be form TR1 you required.
We deal with an application on merit and can't advise as to what's the 'best' option for all hence the recommendation to seek legal advice. Once the 'best' option has been decided upon then our
online guidance can assist re the forms needed.
Answers to your specific Qs re form AP1 follow as they may provide helpful if you then go ahead with a form of transfer
Panel 6 - complete with the full names of those to be added to the register as the new owners eg current surviving owner plus children
Panel 9 - include address for services for each named applicant. It's your choices as to what address(es) you use inc an overseas or email address as the notes explain
Panel 15 - correct
Panel 4 - if the Transfer is from 1 to 4 and no monies changing hands then you should complete the panel to refer to Transfer 1 to 4 - Full market value £X. Then pay the appropriate fee which is payable under Scale 2 on 3/4s of that full value