Solve Tx and Rx separately, using two timers. Start with Tx, that's easier. Start at low baudrate, timer generating interrupts at the barrister rate, and in timer interrupt, toggle a pin as appropriate, shifting out the transmitted value plus the framing bits.
Rx is somewhat more complex - you have to start the timer at the startbit's falling edge, detect that either as external interrupt, or, better but constrained in pins choice, use timer input capture (there a bonus that there is an adjustable filter on it). Then again in the timer interrupt, read the Rx pin and shift in the value.
Should you need higher narrated than the interiors allow, you would need to employ DMA.
JW
