"I remember getting a check in the mail"
Having your hands on the envelope constitutes constructive receipt of the distribution for tax purposes, so it needs to be reported.
If your intent is to roll over the distribution, the IRS provided Revenue Procedure 2016-47 which is new self-certification procedure whereby the custodian receiving the rollover can allow the rollover to be completed after the 60-day deadline if your reason for missing the deadline satisfies one of the listed reasons: https://www.irs.gov/pub/irs-drop/rp-16-47.pdf
In your case, 3.02(2)(b) would apply:
the distribution, having been made in the form of a check, was misplaced and never cashed.
If the check is actually lost at this point, it should be possible to get the issuer to reissue the check.
If you receive a check for a 1099-R retirement distribution, then you have 60 days to deposit that money into another retirement account (indirect rollover) or put it back into the account it came out of. Otherwise the distribution will be taxable and, depending on your age, subject to a penalty.
If you received a check and simply never cashed it, then you need to contact the custodian of the retirement account and discuss the situation with them. To be candid, I'm unsure whether that would be considered a distribution or not, since the funds never actually came out of the account.
If this is what happened, it's possible they could issue a corrected 1099-R to reflect these highly unusual circumstances.
If the curerent plan administrator issue the rollover check directly to the new company's 401k plan around the end of the year but I d requested to re-issue the check in the name of different IRA (still standard 401k not roth IRA) next year, then what will be the tax implication. The original plan administrator already issued the 1099R for 2019 but no financial transaction really happened in 2019. The check is re-issued by the same plan administrator in 2020 several months later and that check went directly to the new 401k plan. Do I need to report that 1099R in 2019 tax return? I think reporting that will cause more complication because the original plan administrator will issue a corrected 1099R for 2020? I hope 60 day rule will not apply since I never got hold of any money during the process, right?
A direct rollover is not subject to the 60-day rollover deadline; the 60-day deadline only applies to indirect rollovers. Unless the 401(k) plan issues a corrected 2019 Form 1099-R to indicate that no distribution occurred in 2019, the IRS is going to expect the rollover to be reported on your 2019 tax return and the 401(k) plan should not be reporting anything about this on a 2020 Form 1099-R. Given that the rollover is nontaxable, there is no reason not to report the rollover on the tax return for the year of the Form 1099-R that reports the distribution.