On my PCB, do I need to keep UART lines away from my 1/4 wave monopole antenna?
Fast edges on the lines will couple, by displacement currents, onto the antenna feed, and perhaps briefly (for the duration of the logic_signal risetimes) overload the first transistor amplifier (the LNA of your receiver).
This overloading will shut down the RF signal (block it) for duration of the risetime.
Can your communication protocol, and your demodulation, and your IF amplifier/filter, and any phase_recovery, handle missing 10nanosecond pieces of the arriving RF energy?
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Let us run some numbers.
Assume two traces; one is logic level (2.5v, 2.5 nanoseconds Trise), other trace is RF trace with 50 ohm impedance (or 150 or 200, or 20, or 40).
Assume the traces are 10cm long and 1mm wide, spaced 1mm from each other. What is the coupling capacitance between them? For an approximation, turn the two  traces on their sides so their area is parallel, assume no fringing, and use this formula:

C = E0 * Er * Area/Distance

which for Eo = 9e-12 farad/meter and Er = 1 (air, FR-4), becomes

C = 9e-12 * Area/Distance

and for our dimensions, we have

C = 9e-12 farad/meter * (1mm * 10cm) /1mm == 9e-12 * 0.1 meter

and we see C between logic and RF traces (10cm long, 1mm apart, 1mm wide) is approximately

C = 9e-12 * 0.1 ~~ 1pF

Now we compute the peak current throu this capacitance, for 2.5v swing and 2.5 nanosecond Rise.

I = C * dV/dT = 1e-12 * 1volt/nanosecond = 1 milliamp

thus the Logic Trace dumps 1 milliamp onto your 50 ohm line, causing 0.05 volts, which is 50,000 microvolts.
How does your receiver recover from that?
Also this much voltage (50milliVolt), is large compared to 0dBm (0.632 volt PP).
Your UART induces -22dBm energy bursts onto the antenna; the FCC likely does not approve of this.
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What to do? Shield the logic trace from the RF trace.

You might actually use a 10cm piece of COAX, with Ground wiring at each end, for the RF path.

Or move the two traces apart, and place a grounded piece of PCB metal between the two; Ground this new piece of metal every 0.5 centimeter.

Or turn off the UART when receiving.