The only correct sentence is It would be appreciated if you would check this information in a timely manner. 'check' means 'make sure that ... is correct'. Neither 'ascertain' nor 'make sure' would fit in this sentence because they both need to be followed by verb phrases. Ascertain what? Make sure about what? To use these other two verbs you'd need a phrase like 'check that this information is correct' or 'ascertain whether this information is correct.'

1) There is a difference in meaning. "To ascertain" means you do not have ANY information yet. "To check" or "to make sure" means you do have information but you are not sure if it is correct, and want it verified. "The road map shows a ferry at Glenora but I can't ascertain what the schedule is." "This 2013 schedule says the ferry runs every fifteen minutes, but I want to check and make sure that it is still accurate." 2) "Ascertain" is a rather formal word. You might use it in a formal business memorandum. But you should know that it isn't plain colloquial speech and sounds a little pompous to me. The ordinary phrase would be "find out." "The road map shows a ferry at Glenora but I can't find out what the schedule is." 3) These three passages are all correct English. (a) means something different from (b) and (c). a) You do not say what percentage of copper is in the ore Wilkins is offering us. It would be appreciated if you would ascertain this percentage. b) The attached memo indicates that the Wilkins ore is 37% copper in sulfide. It would be appreciated if you would check that this is an accurate number. c) The attached memo indicates that the Wilkins ore is 37% copper in sulfide. It would be appreciated if you would make sure that this is an accurate number.