Arrow's abandoned Harley Quinn and Arkham Asylum plot would have been bad news for Batwoman. Back in 2014, Arrow began increasing the prominence of its various Suicide Squad characters, with John Diggle leading Task Force X into the field.
The series also dropped some nods to Harley Quinn, clearly setting up her full Arrowverse debut. Producer Andrew Kreisberg insisted this had only ever been an Easter egg, but it soon became clear the show had been forced to conduct something of a course-correction. Warner Bros. was wary of over-exposing the Suicide Squad characters, and their focus was on the impending Suicide Squad movie. Soon, Deadshot and Amanda Waller were both dead, Task Force X was effectively disbanded, and the Harley Quinn nods were forgotten.
Speaking at the MCM London Comic Convention in 2015, actress Willa Holland - who plays Thea Queen - revealed Arrow writers had big plans for Harley Quinn. According to Holland, Arrow was going to develop its own version of Arkham Asylum, presumably populated by various insane members of Batman's rogues' gallery. "We were going really heavily into that for a minute," she noted. "And then, something must have come down from DC or some higher-up above that said 'No, you must cease and desist because we’re going to make it into a movie and we can’t have anyone spoil that idea.'" Little is known of this arc, but frankly, it's a good thing for Batwoman that it was canceled.
No doubt Arrow would have avoided using the name "Arkham Asylum," but this arc would naturally have caused problems for Batwoman down the road. Arkham is an established part of Batwoman's Gotham, although it's rarely been dominant in the narrative so far, but this would have felt repetitive. There's a risk that Arrow would have mined the concept of its richest story possibilities, leaving precious little left for Batwoman.
Making matters worse, Arrow has always had a tendency of adapting Batman's rogues' gallery, turning them into enemies of Oliver Queen. Characters like Ra's al Ghul, Deathstroke, and Firefly have been reinvented as Green Arrow enemies, and in many cases that means they lost their Gotham history. Had Arrow created its own asylum, it would undoubtedly have used a number of Batman's more psychotic villains. It's not hard to imagine Arrow using prominent Arkham inmates such as Scarecrow, Poison Ivy, or even Bane - thus leaving them off the table for Batwoman. The writers of Batwoman can no doubt breathe a sigh of relief that Arrow had to do a course-correct and that the Arkham plot was dropped.