Is there a ATtiny or similar microcontroller with a UART port?

Go to the Microchip product page:

https://www.microchip.com/ParamChartSearch/chart.aspx?branchID=30047

and select "Show all products" (if not only New/Popular items will be displayed).

And click on the Product Search tab you'll find the microcontroller selector. Using the scrollbar within the page (it's in an odd spot) scroll across to UART and pull the slider down to one and that will show the ATtiny devices with a UART (currently 7 devices).

Most manufacturers and some suppliers have similar search facilities which are normally called a parametric search. Also as mentioned in some comments you can also look at a software / bit-banged serial interface if suitable for your project and then just about any microcontroller will do. That's just implementing the serial protocol in software using regular I/O pins.

The ATtiny2313 has UART. AdaFruit has several projects and tutorials with this chip. The Arduino Platform library can be downloaded from arduino-tiny

Where on the Tiny45/85 it is common to use SoftSerial library to bit bang out the UART. As shown in the Serial communication with the Tiny's tutorial.