"I found a DELTA module 1 distance course that starts in March and ends in June. I'm hoping to work at summer schools at the end of June, so do you think that's enough time to complete the module? Additionally, is it possible to postpone the exam?"
Is it possible to do module 1 and 3 simultaneously? Is it possible to start module 3 before module 1?"
Cambridge holds the Delta exam twice a year, so you could postpone the June exam, but it would have to be for 6 months later. If you want to take the exam in December, just take the prep course that finishes in December. You'll be so tired of it if you study for a year and it's really not necessary. And, you have to register for the exam a few weeks ahead of time and pay a lot of money, so it's not like you can just get stressed out and then decide to postpone it on a whim.
You can do modules 1/3 at the same time, but you'd have to be very organized/extremely self-motivated if you did it by distance. If you do the Delta in person somewhere, you'll often be working on modules 1/2/3 concurrently. You can do the modules in any order, or just do one or two and not the other one(s).