The 3 songs were released within months of each other, but only two were in the chart at the same time.
Frankie Goes To Hollywood released their song "The Power Of Love" late in 1984. It charted the week ending 1st Dec 84 at No3 behind former No1 Chaka Khan (I feel for you) and the No1 at the time Jim Diamond (I should have known better). FGTH stormed to No1 the following week before Band Aid went to No1 the following week (15/12/84). That week Wham entered the chart at No2, pusing Frankie back to 3.
Jennifer Rush's song entered the chart the w/e 29 June 85 and hung around outside the Top 40 until w/e 21 September when it entered the 40 at No36, it then climbed to 15 ---> 2 ---> and No1 (12/10/85) where it stayed for 5 weeks. Her version was responsible for stopping "Take on me" by A-ha from getting to No1.
In the meantime Huey Lewis and the News released a different song with that title which entered the Top 75 on 31st Aug 85 and the Top 40 (07/09/85) at No31, climbed to 19 and peaked at No11 on the w/e 21-09-85. (By coincidence the same week that Jennifer Rush entered the Top 40).