Is there any common country to visit for persons holding UK and Schengen visas?

According to the Mexican consulate in San Francisco, anyone who would otherwise need a visa to visit Mexico is exempt from that requirement if they have a Schengen or UK visa (among others):

effective May 2016 all those foreign nationals, regardless of their nationality, visiting Mexico for tourism, business or transit are NOT required to obtain a Mexican visa IF they hold a valid (non-expired) Visa or Permanent Residence of any of this countries: United States of America, Canada, Japan, United Kingdom or Schengen area (European Union).

So Mexico is one option.

I am not going to confirm these others by checking with the official government sources, but according to Wikipedia, the same is true of

• Albania
• Bosnia and Herzegovina
• Georgia
• North Macedonia
• Montenegro
• Serbia
• Turkey, with some nationalities excluded.

Note that some of these exemptions depend on the fact that the UK is a member state of the European Union, so they may end if the UK leaves the EU. There also seem to be some other countries that allow visa exemptions for only some Schengen and/or UK visa holders, depending on their nationalty. For example, Gibraltar accepts UK visas, and it offers a visa exemption to Schengen visa holders only if they are citizens of China, India, Mongolia, Morocco, or Russia.

Provided OP's UK visa is valid for another five days beyond their intended stay, and OP's friend's Schengen visa is either a) a long-term one, or b) a short-term, multiple-entry one with validity for another five days beyond their intended stay, the Republic of North Macedonia (formerly known as the former Yugoslav Republic of Macedonia) is an option in Europe (the continent).

According to the Ministry of Foreign Affairs, Republic of North Macedonia:

UK visa holders:

Entering the Republic of Macedonia with British, Canadian or U.S. visa

Third country nationals with a valid British, Canadian or U.S. visa with a validity date at least 5 (five) days beyond the intended period of stay in the Republic of Macedonia can stay in the Republic of Macedonia for up to 15 (fifteen) days at any given entry on the territory of the Republic of Macedonia while the total duration of consecutive stays in the Republic of Macedonia must not exceed 3 (three) months in any six-month period calculated from the day of first entry.

Schengen visa holders:

Entry in the Republic of North Macedonia for Schengen Visa Holders

Citizens of the following countries are not required to have entry visa for the Republic of North Macedonia::

[...]

2. Third countries with temporary stay in an EU member country or signatory country of the Schengen Agreement

   may stay up to 15 (fifteen) days upon every entry to the territory of the Republic of North Macedonia and the total amount of the subsequent stays must not be longer than 90 days in any 180-day period.

3. Third countries with permanent stay in an EU member country or signatory country of the Schengen Agreement

   may stay up to 15 (fifteen) days upon every entry to the territory of the Republic of North Macedonia and the total amount of the subsequent stays must not be longer than 90 days in any 180-day period.

4. Third countries with multiple entry short stay Schengen visa type C valid at least 5 (five) days beyond the intended stay in the Republic of North Macedonia.

   may stay up to 15 (fifteen) days upon every entry to the territory of the Republic of North Macedonia and the total amount of the subsequent stays must not be longer than 90 days in any 180-day period.

Note: The Republic of North Macedonia's accession to the EU might change the answer to the above in the future.

Countries in the Middle east

• Qatar(Sure about this)
• UAE

Allow for a transit visa-less access (short duration; 25 hours in my case) when you have a valid Schengen or UK Visa. This might be an option worth exploring!