The divine name YHWH appears only in the Hebrew Old Testament. It appears without vowels in the Hebrew text, and is read with the Hebrew word for ‘Lord’ both by Jews and in traditional Protestant Bibles.
The spelling Yahweh occurs in some Bibles like the Roman Catholic Jerusalem Bible. Yahweh is also how scholars of the Ancient Near East spell and pronounce the name. The Old Testament uses it lots of times (in some English Bibles it’s there whenever the Old Testament says Lᴏʀᴅ or Gᴏᴅ [in small caps]).1
The New Testament does not give a rule to Christians about whether the divine name YHWH should be read when reading the Old Testament, or how it should be read. However it is noticeable that the New Testament writers avoid using the name YHWH even when using the Hebrew title “YHWH Sabbaoth” (which first occurs in 1 Samuel 1:3 at Shiloh) in Romans 9:29 and James 5:4.
Κυριος σαβαωθ = kyrios sabaoth = Lord Sabbaoth, not “Yahweh Sabbaoth”, or “Yahweh of Armies”.
It could be argued that the New Testament is simply following the well-documented Jewish taboo on pronouncing the name YHWH, which originated at least as early as the Maccabean period or earlier. The Greek equivalent of YHWH (“Iao” or “Iabe”) is very rare, found for example in magical inscriptions, not in mainstream Jewish Greek texts or almost all manuscripts of the Greek translation of the Old Testament, the Septuagint.
It might also be argued that this is an authorial choice by the New Testament writers.2
Whatever the reason the New Testament doesn’t use it once, even when translating YHWH Sabaoth, which indicates that Christians then – both converts from Judaism and Gentiles – did not use the name when talking about God or praying to him.
1. See ‘How many times is God called “Yahweh” in the Bible?‘
2. Part of the reason for this might be because, under the New Testament, God has given all power and authority to his son, Jesus (Mat. 28:18), who reigns for him (1 Cor. 15:24-28), so now the focus is on the name of Jesus (Phil. 2:9-11).