If, after passing the final approach fix, weather deteriorates so that the reported values are now below required minima, is it allowed to continue the approach?
You didn't list a jurisdiction so Ill answer for the FAA: Yes you can continue;
If you are flying under part 121 then you can't start the approach but § 121.651 Takeoff and landing weather minimums: IFR: All certificate holders. is going to be the regulation to reference and allows you to continue if you are passed the FAF.
(c) A pilot who has begun the final approach segment of an instrument approach procedure in accordance with paragraph (b) of this section, and after that receives a later weather report indicating below-minimum conditions, may continue the approach to DA/DH or MDA. Upon reaching DA/DH or at MDA, and at any time before the missed approach point, the pilot may continue the approach below DA/DH or MDA if either the requirements of § 91.176 of this chapter, or the following requirements are met: