You need a small change:
DECLARE @xml XML
SET @xml = '
<Columns>
<Column Name="SID2" DataType="String">[{"LastUpdatedUtc":"2021-01-28T11:36:33.407"]</Column>
</Columns>
'
SELECT @xml AS Before
DECLARE @SID NVARCHAR(MAX) = 'SID2';
DECLARE @NewContent NVARCHAR(MAX) = '[{"LastUpdatedUtc":"'+CONVERT(varchar,GETUTCDATE(),126)+'"]';
SET @xml.modify('
replace value of (/Columns/Column[@Name=sql:variable("@SID")]/text())[1]
with sql:variable("@NewContent")
')
SELECT @xml AS After
Results:
Before
<Columns><Column Name="SID2" DataType="String">[{"LastUpdatedUtc":"2021-01-28T11:36:33.407"]</Column></Columns>
After
<Columns><Column Name="SID2" DataType="String">[{"LastUpdatedUtc":"2021-01-28T13:10:24.850"]</Column></Columns>
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