There are a few variations on who sends result letters and when, this is how we do it where I work to give you an idea.
E.g.
Monday: I report a test.
It goes electronically overnight to the health authority/PCT computer, to the screening office.
Tuesday:
The screening office staff (aka call/recall) process the results onto their system and if everything matches, a result letter to the patient is generated and sent out (either business post or hopefully soon 1st class)
Meanwhile in the lab, reports are printed out and sent out to the GPs, some of whom also have electronic links (but not all)
Wednesday/Thursday/Friday - depending on transport/courier/hospital post/royal mail both you and the surgery may have the result letter or the lab report.
When the paper copies are received into the surgery it's generally one person who deals with them. Any straightforward negatives are usually out onto the computer but any report which has anything else on it would go to either the practice nurse, person who deals with cervical screening or quite often to your GP to see and check the management. Your result may be either on it's way to the practice (sometimes they only have deliveries a couple of times a week from the hospital transport) or waiting to be put onto the computer.
This difference in how the reports vs the result letters are sent out is why you have got the result of your test before the practice has.
The screening office is where the invite letters are sent out from, it will usually say 'on behalf of dragonfly's doctors name'.
In some places, individual practices choose to send out invite/smear appointments letters either as well as or instead of the screening office (PCT/HA) letters. They may also choose to send result letters to patients, as well as/instead of.
It does vary which is why it's tricky for me to say for certain who sent you the letter! Under the NHS logo it might say screening services, PCT, call/recall, which would be the centralised office for your area.
It's quite rare for the lab to do the patient letters because of the additional work involved, particularly keeping the patient database up-to-date.
The nhs cervical screening website has more info about results and what they mean. It also has a section with publications and guidance for how the screening programme is run. There is a publication devoted to communication with women about screening, which is quite interesting.
I think it is best for you to speak to the GP, as me trying to work out what the result letter means is guesswork without seeing it, which isn't a good idea for you or me!
However, if it is an inadequate test (rather a abnormal result) then usually we suggest a repeat in 3 months, which as you had your test in late March, would mean you are about due to have one.
Hope you get an explanation!