Electronic – RF switch for multiple antennas

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1. It sounds like the isolation between the antennas isn't great. Maybe the antennas just aren't physically far enough apart, or their wires are too close together, resulting in crosstalk.
2. Switching every few mS probably rules relays out. Here's a diode switcher you could use that only uses common components. The buffer amplifiers should provide approximately 50 ohm output impedance to reduce attenuation. It only needs one GPIO. You could also use jfets for switching but they're less common.

LF RFID at 125 kHz uses magnetic fields to power tags and load modulation to communicate with them. Your reader does not appear to be powerful enough to generate a magnetic field that has sufficient strength. You need to consider a reader that can supply more current to the antenna or some way to boost it.

Look for a "long range" reader (RFID industry term) for a better solution. The small RFID reader boards are "proximity" (industry term) devices that are only designed to work at close range, such as with a handheld.

You do not mention the tag size which also has an effect. A larger one would increase read range. Tiny glass tags have short range even with powerful readers.

Orientation of the tag to the antenna also affects the read range. If your application could allow the tag to be point in any direction you will definitely need a more powerful reader to detect them at a non-optimal orientation.

With FDX, the tag increases and decreases the resistance (shorts) across its internal coil to cause slight changes in the reader antenna voltage to communicate the 1's and 0's (load modulation). A tiny tag does not put a lot of load to an antenna with a weak magnetic field that is 1m away.

Make sure the antenna is properly tuned. The magnetic field strength, current consumption and read range are highest when in tune.

Good luck!