Yes, control and telemetry links are incorporated into the umbilicals. Shuttle launch video, at 9:00, and here are plans for the umbilicals for SLS.
During the launch sequence there's a point where control of the rocket is handed over from the launch control center to the internal systems. For the Shuttle, this starts at T -20 minutes:
Up to this point, control over the countdown and all launch systems resides with a set of computers on the ground, known as the Launch Processing System (LPS). At T-20:00, the Orbiter general purpose computers (GPCs) are loaded with their first set of operations software (OPS1) and formed into a cluster of four GPCs with a fifth online as backup and 'tiebreaker'. From this point on, launch commands pertaining to the vehicle are triggered by the LPS issuing commands to the onboard GPCs and awaiting their report. All backup flight systems are commanded to 'flight' status. Fuel cell thermal conditioning - essentially warmup - begins, and all orbiter cabin vent valves are closed.
The navigation systems come further online; the orbiter's Inertial Measurement Units (IMUs) are initialized. An external bias to compensate for the Earth's rotation is fed into them until just before launch as the onboard IMUs won't show any of the Earth's rotation, due to gravity.
At T -31 seconds, the onboard computers assume full control:
Sequencing and control of the launch process is handed off from the LPS to the GPCs aboard the Orbiter. The GPCs assume control of all launch activity; however, they will still respond to 'hold', 'resume count' and 'recycle' (re-do part of the countdown) commands from the LPS. Note that at and after this point, any 'hold' or anomalous events will cause the countdown to recycle to the T minus 20 minute mark. After this, only one ground command is required to launch.
It makes sense to switch over to radio links after that, but before the launch proper, so you can test the links before you commit to the launch. But I haven't found data on this for the Shuttle.
As for data rates:
During ascent, space shuttle main engine interface unit data are sent to the S-band FM system to be transmitted in real time to the ground. These data also are routed to the operations recorders for non-real-time transmission. On orbit, one of the other six services may be selected and routed through the S-band FM signal processor.
The FM signal processor is commanded to select one of these sources at a time for output to the S-band FM transmitter, which transmits it to the S-band FM return link through the STDN ground station used for launch, lift-off, ascent or landing or the DOD AFSCF ground station. The S-band FM return link transfers real-time SSME data from the engine interface units during launch at 60 kbps each or real-time video or operations recorder dumps of high- or low-data-rate telemetry at 1,024 kbps or one-track dumps of 60-kbps SSME data at 1,024 kbps or payload recorder at 25.5 kbps or 1,024 kbps or payload analog at 300 hertz or 4 MHz or payload digital data at 200 bps or 5 mbps or DOD data at 16 kbps or 256 kbps in real time or 128 kbps or 1,024 kbps of playback.
The S-band PM forward link transfers a high data rate of 72 kilobits per second, consisting of two air-to-ground voice channels at 32 kbps each and one command channel at 8 kbps, two-way Doppler and two-way tone ranging.
The S-band PM forward link transfers a low data rate of 32 kbps, consisting of one air-to-ground voice channel at 24 kbps and one command channel of 8 kbps, two-way Doppler and two-way ranging. The two-way ranging does not operate through the TDRS.
The S-band return link high data rate of 192 kbps consists of two air-to-ground voice channels at 32 kbps each and one telemetry link of 128 kbps, two-way Doppler and two-way ranging. In the high-data-rate mode, incoming signals are directed to a preamplifier before they reach the transponder, and outgoing signals go to the preamplifier after leaving the transponder. The two-way Doppler and two-way ranging are operative only when in view of the NASA STDN ground stations at launch, lift-off, ascent or landing or in view of the Department of Defense AFSCF ground stations. The two-way ranging does not operate with the TDRSS.
The Ku-band system return link consists of channel 1, modes 1 and 2, plus one channel 2, modes 1 and 2, and one channel 3. Channel 1, modes 1 and 2, consists of 192 kbps of operational data (128 kbps of operational data telemetry and payload interleaver plus two air-to-ground voice links at 32 kbps each) plus one of the following selections from channel 2, modes 1 and 2: (1) payload digital data from 16 kbps to 2 Mbps, (2) payload digital data from 16 kbps to 2 Mbps, (3) operations recorder playback from 60 kbps to 1,024 kbps, or (4) payload recorder playback from 25.5 kbps to 1,024 kbps. It also includes one of the following from channel 3: mode 1 attached payload digital data (real-time or playback) from 2 Mbps to 50 Mbps, mode 2 television (color or black and white) composite video, or mode 2 real-time attached payload digital data or payload analog data.
The Ku-band system forward link consists of a mode 1 and 2 through the TDRS in view. Mode 1 consists of 72-kbps data (two air-to-ground voice streams at 32 kbps each and 8 kbps of command), 128-kbps TAGS (used in place of the teleprinter) and 16-kbps synchronization. Mode 2 consists of 72-kbps operational data (two air-to-ground voice streams at 32 kbps each and 8 kbps of command).