I suspect I know the answer to this but I'm curious so I figured I'd ask.
Do any UK networks warn you when you get close to or exhaust/exceed your free minutes?

And I don't mean things like T-Mobile's old Flext tariffs when they'd send a balance each week, I mean current tariffs with a fixed allowance of minutes, where a customer could end up incurring extra charges if they were to go over.
Do any of the networks notify customers in such situations or do they simply allow the customer to carry on and run up a bill?

Three do, you get a text when your allowance is low, they are also always up to date or within a few minutes. So if you download a 10MB file then wait a few moments you'll see it reflect on your balance.

The notifications are important, but of course if the balance system isn't realtime as is only updated every few hours (like it is on some networks), then you don't know you've gone over until it updates.

I am with three and i got a text when just over half my inclusive minutes were used this month which was very useful.

Three send you warning texts. Due to those i managed to keep under my allowance after heavy usage during a month.

I know o2 only sent out internet allowance ones. Been over my minutes a few times without realising. One of the main reasons i moved networks.

Three will actually stop you making out going calls when your allowance is up. Although they only put this service on once. So if you cancel they say they "can't" reactivate it I suspect they can but they make more money if they don't

T-Mobile certainly don't as this situation happened to me last week and i ended up with a phone bill of £191, no joke. I couldn't f*cking believe it!!!! Because the money's taken out by direct debit, as a result it left my bank account overdrawn and i'm now incurring bank charges! Absolutely gutted, extremely annoyed and very pissed off.

giffgaff tell you after each call, internet and texts are unlimited.

You can always contact the network operator or check your account online to find out how much of your allowance is remaining.

If you are on contract and worried about going over your limit ask to see if you can remove the credit limit or reduce it I have with 3. The second I go over my allowance the call cuts out.

3 however do send texts when you are approaching your limit or gone over it.

They should all have to notify users when their spend hits $100,200,300 etc to avoid people (mainly kids) racking up telephone number bills (pun intended) blindly