In ‘The Bells of Saint John’ the Doctor is surprised to get a telephone call from Clara while he is in Cumbria 1207. This isn’t all that strange considering the Doctor later describes the TARDIS as a mobile phone.
Despite his travels people who need the Doctor have often found a way to contact him, whether it be the Brigadier using a space/time telegraph or Winston Churchill ringing him. It has also become a regular thing for the Doctor to modify his companions mobile phones so that they can call home.
It is entirely possible that the TARDIS is capable of broadcasting messages through time but the question would remain how people in other time zones are connecting to the TARDIS.
Time travel offers a variety of possibilities. The helpline could go to an answering machine. At some point in the future the TARDIS checks the answer machine and determines when the call was made. It could then intercept the call at the moment it is made and connect them to the Doctor. Technically this is altering the past but only the TARDIS would be aware of this.
Instead of an answer machine the message could be broadcast to a nearby incarnation of the TARDIS. While the Doctor is off having adventures the TARDIS is routing his phone calls, backwards or forwards to an incarnation who can deal with it (the TARDIS travel logs could help determine which Doctor is free at that moment).
The signal need not travel far, broadcast to the next nearest version of the TARDIS which can continue broadcasting it forward or backward along multiple time ships until it reaches the target.
The signal need not be broadcast at all, simply saved until the Doctor is available. It could wait until the Doctor is bored and looking for an adventure to link him back to the caller in their own timeframe.
It is possible that the Doctor or the Time Lords have set up a time/space communication next work. The Message Cubes used by the 2nd Doctor could have been surpassed by technology that can link to a TARDIS via the vortex.
The important thing in this story is that Clara was given the Doctor’s number by a mysterious woman (River Song?). It just happens that she calls the Doctor just as her life is placed in danger.
Within a campaign, rather than a Time Lord wandering space and time and stumbling across adventure they might give out their contact details so that those who need help can call them.
If present as a helpline then the reason for the call can be quite varied. The player characters could get calls about missing people, mysterious lights in the sky, concerns about strange events in a local area or people in a moment of crisis.
This allows for a wide range of adventures. Once they’ve received a call the player characters have to get to where the call is being made from in time to help them. They can then begin their investigation.
It is up to the player characters how public they wish to make their helpline. They could advertise their services but it might also spread by word of mouth or via contacts. Those who they help could pass the number to others in need.
Creating these links can be a nice way to show the players how their characters have helped others. Their current client could have been referred by someone they have helped in the past and they can find out a little about how their life has improved.
The client could also have been given the number by someone the player characters haven’t helped yet, from their perspective. This can foreshadow later adventures when they recognise someone who referred them in their past who they hadn’t yet met.
The only limitation is that the client must have access to a communication device, whether it be a telephone or futuristic holo-vid. If the player characters are keeping a low profile the caller might not even realise that they are contacting a time traveller.
If you want to concentrate on the communication side, rather than just using it as a means to introduce the plot hook, the player characters could deal with the issues of sending a telephone call through time.
What if they or someone close to them received a dire warning from the future? What if they were told they were going to die or something terrible was going to happen and the call was from themselves? Could they avert the disaster?
What if the call was misdirected and someone gained knowledge of the past or future that they shouldn’t? Would the player characters have to find them before the information was abused?
If technology was being used to co-ordinate these communications what if it was damaged and needed repair? What if it was attacked or stolen? What if someone was able to duplicate it and began co-coordinating an organisation throughout time?
The player characters might be given the responsibility of obtaining stored messages so that the TARDIS can forward them on. This could be a very dangerous task if the storage device was in a war zone or post-apocalyptic setting.
The film ‘Twelve Monkeys’ contains an excellent example of how a message device collected in the future can be distorted or misunderstood. Player characters could find themselves in a similar situation, acting on what they think they heard and learning the truth much later.
Moderation is, as ever, important here. If you do decide to have a helpline within your campaign be careful not to bombard the player characters with phone calls. It is a helpline, not a call centre.
The player characters must have time to breath and turn the phone off if they wish. They do have access to a time machine after all and can respond to missed calls anytime they like.