I think that the meanings of いっかげつ and ひとつき are the same. Here are other いっかげつ or ひとつき 一箇月、1か月、1カ月、1ヶ月、1ヵ月、1ケ月　 ひと月 We can use いっかげつ or ひとつき in a similar way. For example, one or two months have passed, ひと月ふた月経った 1、２か月経った Generally, "箇月" is used with 漢数字(Chinese numeral) like 一箇月, but, か月、カ月、ヶ月、ヵ月、ケ月 are used with 算用数字(Arabic numerals) like 1か月. According to this site, https://landgather.com/month Generally, public officials use か月 and 箇月. According to this site, https://www.koho.or.jp/useful/qa/hyouki/hyouki01.html か月 is the safest way for public use. We can use any of them for private use, but we better not use different types of いっかげつ or ひとつき in a same document. I think that ひとつき sounds older and nostalgic, too

The meaningｓ are the same. They are 99% interchangeable. You can use both words in modern Japanese. The difference is the pronunciation system. 一ヶ月 ＝　一箇月 or 一個月. Ikkagetsu = Ich/ka/getsu, which are all 音読み, Chinese-origin pronunciation ひと月 hitotsuki = hito/tsuki, which are all 訓読み, Japanese-origin pronunciation. This is the same type of difference as that between 一個 and ひとつ 東京に来て一ヶ月が過ぎた。 東京に来てひと月が過ぎた。 ひとつき sounds "softer" than 一ヶ月, as it always goes with other Japanese-origin pronunciation (訓読み）and that of Chinese origin(音読み） You can use whichever you like. It is a matter of preference.

Thank you! どうもありがとうございます!

They are same meaning, but 一月 （ひとつき）is older word .