Yes, you should be given IFR separation. It you are filed IFR on the first leg & execute an instrument approach the controller has to protect for the missed approach which would give you IFR separation. This separation is guaranteed throughout the missed procedure. You should be given a new clearance limit & probably a squawk code & you're still IFR.
If you are IFR and you go missed, you are only allowed the published missed approach (or alternate missed approach if issued by the controller). You can't arbitrarily start a turn "as filed" on your 2nd leg without being cleared for it. Seems simple.
I've had an aircraft doing the exact thing the OP is asking. The published missed approach was "heading 090, climbing to 4000, ...blah blah". The standard IFR climbout at this airport was also "heading 090, climbing to 4000, ...blah blah". I tell the pilot to let me know when he's ready to copy his 2nd leg IFR clearance. He tells me he'll copy it on the go from his IFR approach.
His filed route was a westerly heading, so he flew west on the missed and said he was ready to copy his clearance. I asked him why he was turning west? He said he assumed I was going to clear him "as filed" so he was getting a jump on it so he could copy the clearance while already navigating on course. WAY WRONG. I never had a problem with it before...especially since the Phoenix class B engulfed our class D.
If VFR & doing an approach you are given IFR sep from the moment you are cleared for the approach, with the exception of 500ft vertical instead of 1,000ft. However, unless you are authorized to execute the missed approach the IFR sep stops at the airport.
Your either talking about class B airspace, a Letter of Agreement between an ATC facility and a flight school, or just a common controller practice.
If your VFR doing practice approaches there is no IFR separation requirement. I'll do my best to give you IFR wake turbulence separation when possible, but it's not a requirement. If IFR wake turbulence separation is 4 miles and i only have 3 1/2, then that's all your getting.
Again...there is no requirement to provide IFR separation to VFR practice approaches. This is assuming your outside of class B airspace. IF within class B airspace then you would be correct in stating that there is some sort of separation. That separation would be 1 1/2 miles or 500 feet. I would still try my best to provide you wake turbulence separation...as much as possible.
p.s. ATC phraseology for approving a VFR practice approach is:
PRACTICE APPROACH APPROVED, MAINTAIN VFR, NO SEPARATION SERVICES PROVIDED