It's not difficult to see why the police were instantly suspicious of Luke.
In Judith's first statement, she told police Luke had told her he was "coming up the path on his bike" and that statement wasn't corrected until almost a month later when Judith told police she'd made a mistake - Luke hadn't said that at all, he'd said he was coming up the path "with his dog."
One police officer noted, after taking the missing person details, that Jodi had left her home at tea time "with her boyfriend."
Both officers on the ground and the 999 operator were of the impression that Luke, and Luke alone, was (a) out looking for Jodi and (b) somewhere behind Newbattle High School on a path.
Then the officers on the ground get a shout - the boyfriend's found a body.
So think about it from their perspective - Jodi left home with her boyfriend at teatime, but now he's saying he hasn't seen her all evening. He's going up a path in darkness on a bike and randomly stops to climb over a wall where he "finds" a body. Suspicious, yes?
So much so that they jumped to the conclusion that the other three searchers had arrived after Luke found the body, hence not taking their statements, not asking any of them to go over the wall again to show them where the body was and separating Luke from the others almost immediately. They hadn't a clue that Kelly and Alice had been over the wall, or that Alice had touched the body, or that all four of them had gone down Roan's Dyke path together.
Even the conclusion that "the boyfriend's found a body" was wrong (in terms of the information being passed to them from control.) Luke told the operator they'd found something - she told the officers on the ground, "He won't say what." It was Kelly who dialled 999 a few minutes later and screamed down the phone, "It's a f*cking body."
But the operator(s) appeared to think the calls they were receiving were from the same person - Luke. The operator who took Kelly's call reported he found the caller's attitude odd - he wasn't reacting the way he'd have expected someone who'd just found a body to act - he seemed more annoyed that the police were taking so long to get there.
So, the police were acting on wrong information and then drawing erroneous conclusions from that wrong information, the assumption, from the off, being that there was something definitely not right about "the boyfriend." Had they not believed Jodi left home "with her boyfriend" at tea-time, had they known he wasn't coming up the path on his bike and he wasn't alone - there were four searchers on the path, all of whom were present when Jodi's body was found - and had they taken statements immediately from all four searchers and discovered the double check of the path was suggested by Alice, would they have jumped to the immediate conclusion that Luke was the killer?
I think it would have been less likely.
If they'd received the call "the search party have found a body," might they have been suspicious about Kelly's comment, "I suppose you've been to my house first?" If they'd been told Jodi was supposed to be hanging out in Easthouses/Mayfield and the family search trio were leaving from Mayfield to look for her, might they have thought it odd that they didn't look for her in Easthouses/Mayfield but headed straight for the path? Might they have thought it strange that, although there were four searchers out in two different areas, they were given only one contact number, for the lone searcher coming from Newbattle, and no contact details for the three searchers coming from Mayfield?
Of course, we'll never know, but I do think it helps to understand why they believed what they did in that critical first hour, because it set the direction of the entire investigation.