As @Shmuel says in the comments, the issue is not in writing but in erasing. More specifically, the issue with writing is that it might lead to erasing (in the spirit of do not put a stumbling block in front of the blind).
So first one has to ask if there is a prohibition of erasing the name of God written in English. If yes then it would likely be prohibited to write it as well.
Briefly we learn from the Torah's commandment to destroy idolatry that we should not to destroy the name of God (see Dvarim 12:3-4). However most poskim rule that the name of God written in any language other than Hebrew has no holiness and can be erased. (Shach Y.D. 179:11; Mishnah Berurah 85:10). There should therefore be no halakhic prohibition to write it.
Nevertheless as mentioned here some poskim rule to be extra careful with names of God (e.g., the Aruch Hashulchan CM 27:3). This is likely going beyond the minimum law (similar to the Geonim quoted in commentaries to Nedarim 7) who have a chumra (stringency) to refrain from even saying a divine nickname in vain). See bottom of here.
See here and here for more sources from MiYodeya. And here and here from detailed studies on the topic by R Mordechai Friedman of VBM.
PS. I like the story about Rav Soloveitchik z"l intentionally writing GOD on the board while teaching a class and then just as deliberately and intentionally erasing it, so as to demonstrate by his own example that this was not halakhically a problem. But I couldn't find a source for it.