I absolutely don’t mind, and I understand why this is something that’s not clear! The full timeline isn’t obvious even from the LOTR books — you have to supplement what we know about Éomer from Two Towers with info from Unfinished Tales, which has the only account of the battle where Théodred was killed. But the short answer is that Éomer was dealing with business elsewhere so he wasn’t part of the fighting at the Isen, and he never saw Théodred’s body — only his grave after the fact. Here’s the long answer:
In late February, Théodred was in the Westfold, which was his territory as 2nd Marshal. Scouts alerted him to troops from Isengard preparing to invade from the west. Acting on his own authority — because his dad was Not Well — he went to meet the challenge with Grimbold and their men and also sent a summons to Elfhelm in Edoras asking that he come with relief troops of his own. We don’t know *exactly* what Éomer was doing right then, but he was 3rd Marshal and his jurisdiction was the East-mark. So he had his own stuff going on, and the bulk of his men would be further away from the Isen than Elfhelm and his men were, so Elfhelm was a more natural relief choice.
The First Battle of the Fords of Isen happened on Feb. 25th, and Théodred was killed that night. We’re not told what day he was buried, but we do know that his grave was there, with his banner flying above it, when the Second Battle of the Fords began on March 2nd. So somewhere in between, Elfhelm and Grimbold buried Théodred at the Fords, right where he died. That means this image from the movies, while lovely and moving, is non-canonical — Théodred never got back to Edoras and wasn’t buried there:
News of Théodred’s death came first to Erkenbrand at Helm’s Deep on Feb. 26th, and Erkenbrand sent word on to Edoras. That messenger didn’t make it to Edoras until midday on the 27th, which is the same day Éomer set out to track down the band of orcs (those carrying Merry and Pip, it turns out) that had just been reported in the east. There is some ambiguity as to whether Éomer heard the news of Théodred’s death before he left or not, but he indisputably spent the next few days engrossed in other urgent stuff that would have kept him from grieving or visiting the grave — he had to go track down and slay those orcs; he ran into Aragorn, Legolas and Gimli in the plains and had to decide whether to let them go; and then, when he got back to Edoras, he got thrown in jail for having acted without orders and for letting those foreigners run free in Rohan. He didn’t get out of jail until March 2nd, when Gandalf healed Théoden, and then they were off straight away to Helm’s Deep.
It’s not until after victory has been achieved at Helm’s Deep and all our heroes are on the road to confront Saruman that they pass by the Isen and take notice of the graves that are there. It’s a bit of a sore point for me that NO ONE mentions that Théodred is among the dead (😵😖🤯), but at least Éomer is thinking of him because he is the one to mention the murder of Théodred among Saruman’s biggest crimes when Saruman is trying to sweet talk his way back into Théoden’s good graces once they get to Isengard (“Remember Théodred at the Fords and the grave of Háma in Helm’s Deep!”).
Anywho, obviously the movies chose to mix up the timelines and events a bit for their own dramatic purposes, since they have Éomer arriving at the tail end of the fighting at the Fords, finding Théodred still alive, and bringing him back to Edoras before his death and funeral. It’s a substantial change from the books, but I do really like that they found a way to put Théodred in the movies and to give proper weight and notice to his death! And thanks for the question, I hope the answer was helpful!