Multi Meeter Streaming Serial Output.

[sonicj]

The EEVBlog approved Uni-T UT61E ships with "RS232 Interface Cable & Software"

available from "The city that never sleeps, Hong Kong"  for < $60usd. UT61E via eBay

[saturation]

The Prologix sends all GPIB output as a serial ascii stream with a limited set of user selectable delimiters, its easy to parse and there is a Pascal based freeware programming environs for your tap into speech synthesis; or throw it to Windows narrator for a quick and dirty.  Alas, the OP wanted stand alone speech support, and in single amounts I don't know if its as cost effective since you need an MPU programmed to parse the text.  The Arduino based boards are in the $100+ range just for hardware.

http://toysdownunder.com/arduino-speach-synthesizer-kit.html

Yes, the crop of ersatz talking DMM are of questionable quality as DMM; I read them as $10 type DMM with $30 worth of speech hardware.

For a good DMM with speech, you either take a good one with a interface, Agilent comes to mind over Fluke and Gossen, and parse the output from its USB, but cost effectively, how that's done, I tend to lean to a PC interface for speed, plus you can magnify the LCD screen so the tiny DMM LCD can appear 10x larger.

I've several old GPIB devices.  What I did was get a GPIB-USB converter, so if the OP feels like, using a GPIB device can be an option, but the adapters are not cheap.  However, like my prior post, this will tie you to a PC to do interpretation.

This is the most cost effective one, $150, the Prologix.

Yes, that's a good solution, but a $150 DMM + $150 GPIB interface somehow doesn't sound like the simplest or cheapest solution to this problem. I think the Prologix just uses some FT232 device (or something related like FT2232), so you may be able to find UART signals inside. It can emulate an RS-232 port on the PC side.

see there were several available until 2009.  "Talking multimeter" was the keyword to find other who've done it.

I would be worried about quality. The igi-group one looks similar to cheap, crappy meters, with manual ranging and very little protection circuits. The Omega one looks better (and more expensive), but is not a DMM, more like an ADC. Just a fixed voltage and current ranges I think. Starting from a good DMM and adding a text-to-speech module seems easier than starting with good text-to-speed and adding a bodge-job DMM. Auto-ranging and input protection (eg. mains voltage on ohms range) seem almost mandatory for this application.