They are all correct and they all mean the same, though 'afterward' is a rarely-seen version of the usual 'afterwards'. Perhaps you meant 'afterwards'. [I'm sure you meant 'started', not 'starded'.]
I think 'later' is the most common and 'afterwards' slightly less common. You don't need that second comma in either of these two sentences.
'After' is slightly different.
I bought a phone, but some time after it started to go haywire.
-- means you only bought it after it went haywire!! So why did you buy it?! Even with your second comma it sounds a little ambiguous.
The problem is that we expect 'After' to be followed by something: 'after lunch, 'after 2.00', 'after you went out'. We tend to attach it to whatever comes after it.
Only by re-positioning the commas, or by using dashes --
I bought a phone but, some time after, it started to go haywire.
I bought a phone but - some time after - it started to go haywire.
-- does it make sense.
[Dashes are sometimes considered lazy, so let's use the commas.]
By the way, if it comes at the end of the sentence 'after' works perfectly:
I bought a phone, but it started to go haywire some time after.
So, from most common to least common:
1 I bought a phone, but some time later it started to go haywire.
2 I bought a phone, but some time afterwards it started to go haywire.
3 I bought a phone but, some time after, it started to go haywire.
1 and 2 should only have one comma. 3 must have two where I've put them