What I believe to be a full picutre (suffice to say, it would be prohibitively complex):
I believe you need to be sponsored by a company approved/accredited by the Sentinel scheme, who are the admininstrators of PTS qualifications. So, you couldn't just set-up a company purely the purpsoes of sponsoring your own PTS training.
If you received said sponsorship, then yes, you could pay for your own PTS training and receive it.
However, if you intended to access track-side by yourself, you would also need to have the Controller of Site Safety (COSS) and Individual Working Alone (IWA) competancies on your PTS record. These require additional training and would very unlikely be given to an individual who hasn't demonstrated an ability to use the standard PTS qualification safely first, for a number of months.
If you received all of the above, you're a COSS & IWA and have completed and submitted the required Safe System of Work (SSOW) paperwork to your PTS sponsor, then you would still have another hurdle to overcome. You would need a Network Rail (NR) padlock key to physically gain access to the infrastructure. By the book you'd been to be given one of these by Network Rail to grant you access. All of the above demonstrates that you'll be safe on the Railway; the latter is your actual legal means of gaining access to NR's track-side property. Anything else would be tresspassing.
Finally, even if you did have all of that, none of it entitles you access to operational areas such as signal boxes. To do the latter you'd need to receive approval from NR. They're unlikely to grant external parties access for photographic or heritage-interest reasons outside of pre-arranged and sporadic visits with railway interest groups (for which, none of the above would be required anyway).