The British company said the search for a chairman is progressing well and that Sharon Baylay will still lead as acting chair in the interim period.
Osborne, who has worked at Debenhams, John Lewis and a handful of other well-known UK retailers, joined the company in November as its CFO.
"I look forward to leading the business through this critical period of change and seeing the benefits of our plan come to fruition," Osborne said on Monday.
The statement came in after the company shut businesses earlier this month due to the coronavirus pandemic.
Osborne's predecessor Page left in December after multiple profit warnings that had come on the back of tough retail conditions, including subdued spending by consumers and steep costs.
The British retailer also suffered a string of management-related issues, given the misconduct allegations in 2018 against founder and top shareholder Ray Kelvin, who has denied them but stepped down later.
(Reporting by Tanishaa Nadkar and Muvija M in Bengaluru; Editing by Bernard Orr and Sherry Jacob-Phillips)