The Doppler effect is of concern when communicating with satellites that move at kilometres per second speeds. Bikes don't move fast enough to give significant frequency shifts. Also, in an MRT, both parties don't move relative to each other (they go at the same speed in the same direction), so this is a non-issue.
One thing that affects the performance of cheap consumer radios is of course that the receivers are usually crap - they may go pretty deaf in the presence of signals on other frequencies. Plus, a lot of cheap electrical appliances sold here are effective radio jammers (IDA is probably too busy preventing responsible people from using radios than taking enforcement action against trashy electronics importers/manufacturers).
Also, if the radios use squelch (and preset squelch levels are generally fairly high to mask the receiver flaws), the radios will not unmute even if there still is a usable signal. Some radios may have a "monitor" or "unsquelch" button, one could try that to see whether there's a signal.
I'm not sure if tone-coded squelch (often advertised as "privacy codes") is legal in Singapore, but a lot of such radios are advertised as having more "channels", which is an outright lie (like so much what you read in marketing & advertising). Tone-coded squelch helps to separate different parties using the same frequency, but only one party can use the frequency at any one time. If two or more parties transmit on the same channel (in the erronous belief that the channel is free, because they didn't hear anything though their tone squelch), they unknowingly jam each other. This problem can also be recognized if the radio has a "monitor" button, which lets you hear whether there are other parties before transmitting.
