AWS D1.1 is about 400 pages long and costs about $350. There is no condensed version. Purchasing of the welding codes appropriate to your applications should be part of your standard cost of doing business as much as the purchasing of your tools and raw materials.
If the weld is convex, you're okay in terms of effect on weld throat size as calculated from leg size, though perhaps the profile might be problematic in terms of stress concentrations. AWS D1.1 isn't appropriate for very thin material. You could try AWS D1.3, the sheet code, but it sounds like you might need a pressure vessel code.
If by "not knowing where the limits of the pre-welded parts were" you mean you might have unknown fitup gap, I don't know how to help you there. The weld size needs to be increased by the amount of gap, but you have no way of checking this once the part is already welded. Sounds like you really need to set up a proper welding program including quality control and inspection at all the appropriate stages. Hire a welding engineer, or at least bring in a temporary consultant. If failure of this weld might endanger human safety, your company is acting irresponsibly by welding without really knowing what you're doing.
If you're concerned about the 0.125" vs. 0.06", oversize welds are generally not disallowed for structural reasons. They can be problematic if the welding melted through the base metal, especially with double-sided fillet welds in which the long shape of the combined nugget can lead to cracking during cooling. They can give a false sense of security if the base metal strength is less than that of the oversized weld. At least in D1.1, for materials thicker than your application, fillet welds in lap joints are required to be 1/16" smaller than the material thickness so that the the unmelted edge of the material can still be seen, becaus a melted edge might disguise an undersized fillet weld.
All in all, if someone gave you a 1/8" weld instead of a 1/16" weld, that in itself isn't a problem, but if you have an unknown fitup gap that might be reducing that 1/8" weld to something less than 1/16", you could be in trouble.
Hire a welding engineer.
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Eng-Tips guidelines: faq731-376