Ted, along with Dirk Cresswell and Gornuk, spend the fall of 1997 and the spring of 1998 on the run. Sometime in March, they are all captured and killed (DH22).
Timeline Notes
While the dates of events in Deathly Hallows are difficult to determine, it is possible to narrow the death of Ted Tonks down to March, since immediately after those deaths are announced on Potterwatch Harry says Voldemort's name and he, Ron, and Hermione are captured by Snatchers. This sets off a chain of events culminating in the invasion of Gringotts, which we know is about eight weeks later.
They remained shut in the cupboardlike room for hours at a time. Slowly the days stretched into weeks. There was problem after problem to overcome, not least of which was that their store of Polyjuice Potion was greatly depleted.
Then a bit later:
Harry felt responsible: It was, after all, he who had insisted that the goblin remain at Shell Cottage so that he could question him; his fault that the whole Weasley family had been driven into hiding, that Bill, Fred, George, and Mr. Weasley could no longer work. “I’m sorry,” he told Fleur, one blustery April evening as he helped her prepare dinner. “I never meant you to have to deal with all of this.”
No further information is given in the text about the dates except that Harry tells Fleur that they won't be with them much longer. The Gingotts affair took place the day before the Battle of Hogwarts, which would be May 1.