The IRS has posted a statement on this:
The IRS is continuing to closely monitor the pending legislation in Congress, and we are taking the initial steps to prepare guidance on withholding for 2018. We anticipate issuing the initial withholding guidance (Notice 1036) in January reflecting the new legislation, which would allow taxpayers to begin seeing the benefits of the change as early as February. The IRS will be working closely with the nation's payroll and tax professional community during this process.
Mostly, all an employer or payroll processor would need to do would be to update their withholding tables, so it would be a fairly simple update. That IRS guidance would presumably include a deadline for implementing it; I suspect if you wait a few days more information will come out regarding the specifics.
Various sources, including major payroll processor APS, have indicated that it's also possible that form W-4 will need to be updated (since exemptions no longer exist per se). This would be a more significant change, which could take longer to implement both in terms of the IRS needing time to decide how to implement the rules, and for payroll processors (especially smaller ones) to implement the rule changes.
Edit: In the Conference Report, I noticed something interesting:
The conference agreement provides that the Secretary may administer the withholding rules under section 3402 for taxable years beginning before January 1, 2019, without regard to the amendments made under this provision. Thus, at the Secretary’s discretion, wage withholding rules may remain the same as under present law for 2018.
The IRS seems to be saying they will go ahead and administer new withholding rules, which this suggests they're able to do, but not required until 2019. It's possible the new rules will be allowable in 2018 but required in 2019, or something else.
The specifics from the law, section 11041 (the section that removes the personal exemption):
(f) Effective Date.—
(1) IN GENERAL.—Except as provided in paragraph (2), the amendments made by this section shall apply to taxable years beginning after December 31, 2017.
(2) WAGE WITHHOLDING.—The Secretary of the Treasury may administer section 3402 for taxable years beginning before January 1, 2019, without regard to the amendments made by subsections (a) and (c).
(a) is the removal of the personal exemption, and (c) is the instructions for modifying the W-4, more or less. (The W-4 will still be basically as it is, just with "allowances" instead of "exemptions", if I read it right. So the difference will just be in how the formulas are applied to those numbers, and perhaps some instructions, though that could be enough to mean you need to file a new one in any event with slightly different allowances if you have a nonstanard situation).