Hi
If you are UK resident and UK taxpayer then you are liable to pay capital gains on your world wide assets, so once you settle your tax in bulgaria (I think there is none as they treat you more favourably), you need to declare your capital gain in your UK Tax return and pay tax if any due.
Due to double taxation agreement between bulgaria and UK, you would be allowed to offset any bulgarian taxes paid off the UK liability.
The amount you have to pay depends on what the gain is.
you are allowed to make £10,000 without paying any capital gains, then anything over this is taxed at 18% for basic rate tax payers or 28% for higher rate tax payers. If the property deed/notary is in joint names, then you are allowed 2x 10,000 allowance, allowing you to make £20,000 profit tax free.
The profit is the difference between the price on the deeds and the price when sold. So lets say deeds say £50K and you sold for £70K, then allowing £10K tax free gain, you pay 18% of the other £10K gain = £1,800
If you currently are a higher rate tax payer you will owe tax mane £2,800
There are various rules and ways to reduce the gain I believe (renting out the apartment gives you more allowance, declaring the apartment as your primary principle residence for a short period), but there are implications of doing these and you should consult with a tax adviser to understand whats possible and whats not advisable!
Now if you dont tell the UK tax man about the gain you might this think you can avoide UK capital gain, but looks at how off shore bank accounts which were once safe havens are having to declare and tell revenue about money you may have "hidden", don't think it will be too long before our greedy government forces other countries or banks in Bulgaria to inform them of sales or monies held in foreign accounts...
Hope this helps a little, but its a huge area and not simple Question & Answer
Paddy