No one can add the dining plan at check-in.
DVC members pay for it at check-in, but it must be added to the reservation at least 2 days prior.
Tickets are a separate purchase for DVC members - the tickets are not tied to the reservation at all. So DVC members could purchase tickets at check-in. DVC members do not have to purchase tickets to get the dining plan.
Package guests must add it at least 3 days prior to check-in. Packages must be paid in full 45 days prior to check in - so if you add
DDP to your package 4 days before you check in, you must pay for it when you add it.
There are a couple of reports that annual passholder room/dining package bookings have to have it added up to 6 days before check-in, but there is conflicting information on that one. It may be 3 days also.
If you are not an annual passholder or a DVC member, you must purchase a package including your resort room and at least 1 day base park tickets (you may purchase any type or length of ticket, but you must purchase some sort of ticket in conjunction with the package) for every member of your party age 3 or over in order to also get the dining plan.
(I do not know what the rules are for booking from the UK, or if they are different from the US booking rules for packages)