The Health Insurance for Foreigners is something to keep immigration happy (D visa, etc.). So if you're making a visa/residence application, this is the right option. However, it's not really of much use to replace proper private health insurance (i.e. it covers emergencies/accidents or repatriation).
I'd recommend getting your EHIC/GHIC if you haven't already. Again, not a replacement for a UK private insurance, but worth having (especially as it's free).
Bulgaria has a public health system similar to the UK's NHS. And it has private health insurance similar to BUPA.
If there are health issues to consider, and you currently have private health insurance, then you probably should look at getting the same type of cover here. There are several companies that offer it, and I doubt it's more expensive than BUPA.
Once you've got your D visa and residence sorted, you could try to register in the public system (by making the appropriate social security contributions, much like one does in the UK with NI). This would get you free or nearly-free treatment in public hospitals.
If you're enrolled in the public system and paying your "stamp", then the private insurance mentioned above would cost even less. (Because most private hospitals here are affiliated with the public system and get some reimbursement, so you only pay the difference.)
This public/private cooperation is a wonderful aspect of the Bulgarian system. The UK public/private systems are separate. So if you want private treatment, you pay the private price. (With the exception that sometimes the NHS when faced with a huge backlog will pay for you to receive an op at a private hospital. But you have no control over this.) In the Bulgarian system it's your choice: for any treatment/operation, you can elect to use public hospitals or private hospitals.
I'm enrolled in the public system (I pay 28 lev as unemployed), but I've never used a public clinic or hospital. I've been in to take a look, and they are as crowded as in the UK. I've exclusively used private hospitals which are (at least here in Plovdiv) very new and rarely busy (and surprisingly inexpensive).
I've got the public cover, and if I were to need an operation, I would definitely use it. But so far (6 years) everything I've asked for has been pretty cheap, even at full price... and I can have it RIGHT NOW, dammit! :-)
Examples...
Consultation with orthopedic surgeon 70 lv.
Consultation with a cardiologist (with a bunch of heart tests) 150 lv
Consultation with a dermatologist 100 lv
Consultation with my GP 25 lv (5 lv if you are in the public system)
X-ray maybe 30-50 lv maybe now, maybe in an hour or so
MRI around 300 lv maybe later today/tomorrow
If you go for the GP consultation, and he thinks you should have an X-ray or MRI (or whatever) then he can make a referral, and you can have it go through the public system. Then it might be 20 lv for the MRI, and you'll get an appointment in a month or so.
Private blood test lab (e.g. Ramus) will do a complete panel for 50-150 lv (depending exactly what you ask for)
Private dentist regular check-up and cleaning, maybe 150 lv (with a dentist, not a hygenist, and using the fancy carbonate spray system)
Hip replacement (pretty serious op), maybe 15,000 UKP in a BUPA hospital... maybe 1,500 euros in a private hospital here (after the standard public payment)