With an Advanced Purchase UK train ticket, what happens if your train is cancelled?

As Gagravarr said, you are entitled to a full refund on the ticket if you choose not to travel as a result of the disruption. However, if you do choose to travel, here's what will likely happen.

Firstly I would be remiss if I did not mention that on an Advance Purchase ticket, it isn't necessarily the case that every leg of your journey is mandatory. Only trains that are marked as having reservations available in the system/timetables[0] are mandatory; if your itinerary involves a train that does not have reservations (in which case your ticket/reservation coupons/eTicket themselves will not mention this train), you may freely use alternative trains which also don't have reservations. This usually applies only to minor connecting services off the main part of the journey, however.

So what should you do if a mandatory portion of your journey is disrupted? If there are staff at the station, first of all you should ask them for advice. If there are general customer service staff (with a helpdesk) it's best to ask them; else the ticket office staff might be able to help you. Gate-line staff are less likely to be able to find out details but you can try asking them. If it's a small station without any staff, you can try using a help point if one is provided but I wouldn't count on that helping much. If the train is scheduled to stop at the station for a few minutes, and it has a guard/conductor/train manager/whatever the operator calls it in your local area, you could try asking them.

In general, if it's small/localised disruption limited to one train only, and this train is cancelled or severely delayed, you will generally be allowed to travel on the next available service operated by the same company. This is what is recommended by internal industry guidelines and is widely followed. If you find out far enough in advance to get to the station in time, staff will usually be kind and let you travel on the previous service too, though this is not guaranteed. If your journey has multiple legs with reservations on multiple different operators, and a delay or cancellation on a previous service meant you will miss a connection in the later part of your journey, you will be entitled to use the next available service run by the same operator as in your original itinerary — this particular right is guaranteed by the National Rail Conditions of Travel and so there is no need to ask staff for permission:

9.4 Where you are using a Ticket valid on a specific train service or train services (such as an 'advance' Ticket) and you miss a service because a previous connecting train service was delayed, you will be able to travel on the next train service provided by the Train Company with whom you were booked without penalty.

However, if the disruption is more wide-ranging, this is where you have to start relying on railway politics being on your side. If an entire route is disrupted, and you wish to take an alternative route (especially one run by another operator), this is subject to temporary ticket acceptance agreements being made between the operators in question. This is by no means a guarantee, and failure to abide by these rules might leave you treated as if you had no ticket at all, so for this reason it is very important at this stage to contact staff and check what agreements are in place. Station staff should hopefully be able to tell you whether agreements are in place for you to take an alternative route, or if you just have to wait for disruption on your original route to end. If they can't tell you this, it is worth contacting the operator in question on social media — most train operators are very responsive on Twitter. Tell them the details of your journey and ask about ticket acceptance on the alternative route you wish to use. Hopefully you'll get the answer you want in quick enough to make use of it!

If you don't get a response on social media, and the staff at the station don't know (or there aren't any), the only thing left to do is to try asking the guard of the alternative train you wish to catch, if that train has a guard. The guard will almost certainly be able to find out for you (using internal industry systems) whether or not ticket acceptance is in place, so flag them down on the platform and ask your question, being mindful that they will not wish to delay their train so you should be as brief as possible in your explanation.

[0] Note that this does not necessarily mean that the train actually has seat reservations; many trains that are restricted for Advance Purchase tickets don't have seat reservations. They get around this by issuing a "counted place" reservation coupon, where you don't get a specific assigned seat but DO get a "reservation" on that train.