As long as any handshake signals doesn't block the transfer of serial data, you should be able to read the characters one-by-one without a problem. The microcontroller will still be able to send a large number of characters into the very large FIFO available in USB-to-serial adapters. So a serial read would not result in a USB turn-around event stalling the USB channel and limiting the bandwidth.

USB-to-serial adapters have huge FIFO sizes just because of the high latencies with USB.

But the problem here is that people spends time trying to suggest different solutions, while we haven't even received a good explanaition what the problem is. We do not know if there is a bandwidth problem. Or a sampling rate problem. Or a task-switch problem. Or a problem that a sample gets time-stamped first when picked up from the serial port (after an unknown wait in the FIFO). Or if the OP have other problems.

So in the end, we should stop wasting our time until the OP do give enough information to make it meaningful to restart the thread.