From the release notes over at atari-forum and here, and other sources including contributors in this thread, I've found the following cores are mentioned as supporting ADC.
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Spectrum (verified)
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Amstrad
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Apple II
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ADCTest
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(Unofficial) Fabio Belavenuto's MSX1
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(Newer, now official) molekula's MSX1
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Acorn Electron
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Lynx 48 and 96
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Oric
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CoCo2
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AliceMC10
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SVI-328
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C64
Similarly, I have not found a list of cores that support the RTC, and from the release notes and other sources I have found these:
(verified) Menu (on top of the panels)
(verified) Minimig (Amiga) (Utilities > Clock)
(verified) QL (Sinclair QL) (PRINT DATE$)
(verified) Archie (Acorn Archimedes) (Apps > Alarm)
(verified) NeoGeo ([DIP] Settings: ON > BOOK KEEPING > CABINET/COIN, check date of log entries)
(verified) SNES (Daikaijuu Monogatari II, Tengai Makyou Zero - once set up, you need to reload the core for the clock to take the real time)
(verified) GBA (checked with Pokémon Sapphire, wall clock in first room)
(verified) Gameboy (GB and GBC) (Pokémon Gold: set clock, go one room right, set day of week, save, restart; clock is in the menu screen)
(verified) ao486 (PC) (DATE on MS-DOS prompt)
(verified) X68000 (DATE on HUMAN68K prompt)
(verified) PC8801mk2SR (PC-88) (PRINT DATE$, shows year 85 but month and day are OK, and PRINT TIME$ shows correct time)
(verified) Zet98 (PC-98) (DATE on MS-DOS prompt)
(verified) MSX ("MSX3") (DATE on MSX-DOS prompt)
(verified) BBCMicro (PRINT TIME$ in BBC Master mode)
(verified) TSConf (top of Wild Commander screen, also MaxiClock plugin)
(verified) ZXNext (ZX Spectrum Next) (main menu in Next mode)
(verified) C64 (boot GEOS and use it to run the CP Clock application)
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C64 TOD clock in CIA chip (mentioned in the release notes: viewtopic.php?p=25021#p25021 - no idea how to test it - possibly not battery backed)
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CoCo3 (according to viewtopic.php?p=42665#p42665)
The entries with (verified) are the ones that I have personally verified, indicating how they were checked.
Any help making this list more complete will be appreciated.