You had better look for a way to be in Japan longer, hopefully for at least 2 more days.
If you don't speak Japanese well, it is preferable that a reliable native Japanese speaker can help you.
Travelled from Kyoto via the Shinkansen. I reached Osaka station and caught a cab to my hotel Was that station really not Shin-Osaka Station but Osaka Station?
(Shinkansen goes not via Osaka Station.)
Haven't you got anything such as a receipt which shows the name (and phone number) of the taxi operator?
Osaka Taxi Center can inform taxi operators who have offices in a specific zone including Osaka City about articles left in their taxi cars.
Your phone number is necessary for this foundation to contact you about your lost property.
(Public interest incorporated foundation) Osaka Taxi Center
- Web form in Japanese for notification of lost articles
@
http://www.osaka-tc.or.jp/inqlost.php- Phone numbers for notification of lost articles, complaints
@06-6933-5618, -5619 (#)
@
http://www.osaka-tc.or.jp/pdf/receipttel.pdf@: The phone staff is ready from 9 to 17 on weekdays and from 9 to 12 on Saturdays.
@: The 21st of March 2018 will be a public holiday in Japan, not a weekday.
I have made a police report at the nearby police post. It may take a few days from when the police receives submitted lost property until the police disclose on the website the information about it.
I suppose you know one or a few facts regarding your lost property which other people either cannot know or probably don't know (such as how to log in the system of your laptop computer when it is locked).
Keep such facts hidden so that the police or the operator of your taxi can believe you as the person who actually left the articles.
Osaka Prefectural Police
- Website in Japanese for search for lost articles
http://www.otoshimono.police.pref.osaka.jp/(#) To make an international phone call at the number, you omit the first 0 and add Japan's country code 81.
Japan Guide
- How to dial
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https://www.japan-guide.com/e/e2223_how.html@: See "From Abroad to Japan" section.