What screw size can I go to when re-tapping an M6 bore?
Tap size chart and Drill Size Chart are the useful search terms. Most any useful tap size chart will have the drill size listed. A drill size chart will help you sort out the relative sizes of drills in fractional inch, number, letter, and metric sizes. Taps you can get at the hardware store may simplify the process by being packaged with the correct tap drill. Ordering on line from a &quot;real metalwork supplier&quot; (often less expensive, and far more selection) you'll likely have to order them separately.
What will work depends just how stripped out it is.
1/4-28 or 1/4-20 are the &quot;next size up&quot; with 1/4-28 offering a bit more allowance for hosed threads since it's a finer thread (and thus a bigger tap drill - #3 = 5.4102mm) but if the holes are stripped all the way to 6mm that's still a bit precarious. 1/4-20 uses a #7 tap drill = 5.1054mm)
The rather unusual but extant M7 is the first size that will work if the holes are ripped out to 6mm - 7x1 uses a 6mm tap drill, 7x0.75 uses a 6.2mm tap drill.
Next up from that is 5/16-18 (F tap drill = 6.5278mm) or -24 (I tap drill = 6.9088mm)

UNC = Unified National Coarse
UNF = Unified National Fine
UNEF = Unified National Extra-Fine

SAE (society of automotive engineers) is an older standard that has been incorporated into the Unified Thread Standard.
You can also repair the threads to M6x1 by using a Heli-Coil® thread repair kit (or possibly generic knockoffs if their patents have expired) where the hole is drilled out larger, larger threads are made in the larger hole, and a Helical Coil that fits in the new threads on the outside and has M6x1 threads on the inside is inserted in the hole. [I have used this product rather extensively, but am not associated in any other way.]