(OMG I skipped writing in my diary for three days....) (OMG I skipped writing in my diary for three days....)

(OMG I skipped writing in my diary for three days....) (OMG I skipped writing in my diary for three days....)

or: I skipped writing a diary entry for three days

These days, I've been listening to a podcast called Luke's English pPodcast to learn English expressions and get used to the sound of English. These days, I've been listening to a podcast called Luke's English Podcast to learn English expressions and get used to the sound of English.

"Luke's English Podcast" is the name of a podcast, so all of the words should be properly capitalised.

These days, I've been listening to a podcast called Luke's English pPodcast to learn English expressions and get used to the sound of English. These days, I've been listening to a podcast called Luke's English Podcast to learn English expressions and get used to the sound of English.

Assuming that the name is "Luke's English Podcast" ... if so, all (major) words in title are capitalized in English.

Teacher Luke is British. Teacher Luke is British.

The teacher, Luke, is British. Sounds more natural. Teacher Luke is fine if that is what he calls himself and not just his proffession + name

The teacher, Luke is British. The teacher, Luke is British.

I could not hear them apart difference between American and British English, but recently I can. I could not hear the difference between American and British English, but recently I can.

Previously, I could not hear them apart difference between American and British English, but recently I canhave been able to. Previously, I could not hear the difference between American and British English, but recently I have been able to.

(1) This entire diary entry is written in the past tense, so there is a need to establish which events happened first. The addition of "previously" here makes it clear that you're referring to a time further in the past. (2) "Recently" actually refers to a time in the PAST that is very close to the present. Therefore, we generally do not use the simple present tense ("can") with it. Instead, we typically use what are known as the "present perfect tense" and "simple past tense".

Previously, I could not hear them apart difference between American and British English, but recently I can. Previously, I could not hear the difference between American and British English, but recently I can.

I could not heartell them apart betweenthe American and British English, but recentlyaccents, but now I can. I could not tell them apart the American and British accents, but now I can.

I could not hear them apartn't tell the difference between American and British English, but recently I can've been able to. I couldn't tell the difference between American and British English, but recently I've been able to.

or: I couldn't distinguish between... I can = できる I've been able to / I've become able to = できるようになった

This is just my opinion, but I feel that British English is easier to pronounce for Japanese people. This is just my opinion, but I feel that British English is easier to pronounce for Japanese people.

I felt it was difficult to use ithem properly, catching their nuances. I felt it was difficult to use them properly, catching their nuances.

Modal verbs is plural so we refer to "them" rather than "it"

I felt it was difficult to use ithem properly, catching their nuances. I felt it was difficult to use them properly, catching their nuances.

"Modal verbs" is plural, so we use the plural "them" here.

I felt it was difficult to use ithem properly, catching their nuances. I felt it was difficult to use them properly, catching their nuances.

I felt it was difficult to use it properly, catch and understanding their nuances. I felt it was difficult to use it properly and understanding their nuances.

I felt it was difficult to use it properly, catching (or noticing) their nuances. I felt it was difficult to use it properly, catching (or noticing) their nuances.