1) It is to keep abreast of the times that is the aim of studying English.
2) It is the aim of studying English that is to keep abreast of the times.
OK, one could argue that 1) is grammatical but it is very awkward due to the semantic content: keep abreast of something and studying a language. One does not study a language to "keep abreast of the times". One reads newspapers or watches news/cultural programs to do that, for example.
They don't work because the phrase "studying English" cannot have an aim. A person can have an aim in studying English. Likewise, "reading newspapers" cannot have an aim. Only a person doing something can have an aim.
Rewrite:
1) It is to keep abreast of the times that is the [my, your, our] aim in reading newspapers. [grammatical and makes sense]
2) It is [my, your, our] aim in reading newspapers to keep abreast of the times. If you put in THAT, it becomes nonsensical.