I am assuming you are looking at Elaran’s translation; if not then let me know what translation you were referring to so I can comment on it specifically.
First thing to note is that you are misattributing echuidathar to “shall spring”. The line Od i lith naur echuidathar is equivalent to the line “From the ashes a fire shall be woken” in the original poem; both are the 5th line of the respective texts.
Echuidathar comes from the verb echuida- “to waken, rouse”. It is extrapolated from the attested (i.e. written down by Tolkien) primitive Elvish verb ✶kuitā-, a causative formation from the root KUY “awake, live” (i.e. it is derived as lit. “cause to awake” > “to awaken”). It is etymologically related to S. echui that you mentioned.
You correctly observed that the -(a)tha suffix is used to construct the future tense; thus echuida- > echuidatha “will awaken”. You are also correct in observing that echuidathar is the plural form thereof.
The reason that the singular noun naur “fire” is followed by a plural verb is because naur is not the subject. Notice that the original line reads “a fire shall be woken”, which is a passive construction - something/someone will awaken the fire. The way Sindarin handles such phrases is by saying “a fire (they) shall awaken”, with an unexpressed plural subject (“they”). This is based on an example attested in the King’s Letter: i sennui Panthael estathar aen “who should be called Fullwise”, lit. “that rather Fullwise (they) will call him should” - a bit complicated to parse through the literal translation admittedly.
The same passive construction is also seen in cýrathar in the 7th line.