Legal, not an issue in a fictional setting, but practically, no.
The entire concept of cell tower infrastructure technology is to be integrated into the entire cell network system. The cell tower has to be part of a network, backed up by a computer, such that the central network reads your smart phone location from several towers, decides which tower is the strongest signal, and switches to that tower. With only one tower, this switching is not functional, but you still need the central network computer to bridge the smart phone to the Internet.
Unless this cell tower was 'sponsored' by a major cell phone provider, a very major part of the technology infrastructure would be missing. The tower might be there, and fully functional, but there would be no connectivity with the rest of the needed infrastructure.
Unless, of course, the person had enough resources and money to set up their own entire cell network infrastructure, including central computer and interface with the Internet, but in that case there would be a lot of better alternatives available. It would be over-kill for only one tower and one phone.
With respect to the legality, cell towers use an EM (or radio) frequency, which is highly regulated in the UK. Only licence holders can use any portion of the EM spectrum. Periodically, the various EM bands are auctioned off as 'spectrum' to providers with very deep pockets, for their exclusive rights. Even citizens band radios are limited in power when used on public EM bands, unless the person has a licence. So at a minimum, in order to be legal the person would have to obtain a licence to access a particular frequency for the tower, and this would certainly conflict with the already-issued licences of the existing providers.
Even home WiFi wireless routers are restricted by law in their power output, and restricted to the frequencies approved for and licensed to the manufacturer.
It is not the cell tower per say that is the legal issue, it is the particular frequency that it uses that is highly regulated.