The standard advice, should one need to do a tight turn, is to fly as slowly as possible, because that reduces the turn radius.
But if one is willing to trade energy for making the turn inclined (i.e. do a chandelle; perhaps a more aggressive version of the standard 30-deg-bank FAA CPL chandelle) then extra speed is obviously necessary.
There are obvious aerobatic methods of doing a very tight turn, involving a climb with a stall turn at the top, but I am thinking of what would be best for a pilot not trained in aerobatics, should he/she need to do a very tight 180.
I've performed that turn manoeuver several times without any training in aerobatics and found it to be very easy and effective. At normal cruise speed, pull up until you get close to stall speed, then full rudder and aileron in the desired direction and push down to gain speed. Doesn't need much space at all but you lose a couple of 100ft.
I don't think there is anything special about it, neither high g loads nor unusual attitudes. I would always choose this manoeuver and not try a slow regular turn because if you realize the turn radius is too wide, you increase bank and risk a stall / spin.
However, you have to be rather stupid to trap yourself in a box canyon these days. I remember flying in the Alps with a C172 spam can without sensible GPS where on each turn one was hoping that the finger was pointing to the right spot on the map and that there was actually a valley behind that corner.
Welsh valley or grand-canyon?
My personal vote would be to keep the speed unless there's a low cloudbase, as most British valleys open out at the top and the extra altitude will give you more room regardless of the method used to make the turn.
My dilemma is how to practice these. I'm not daft, but as most of my flying is in Wales the prospect of flying up a valley is of concern. I have very little intuition of how large a turning circle my aircraft has, relative to the terrain. I have done about 5 hammerhead turns on a Super-Decathlon and would be fairly confident of being able to manage one unsupervised on that aircraft, but less certain about bringing one off on the C152 that I normally fly.
I'll look forward to Pilot-Dar's views.
A quick back of envelope calculation shows the turn diameter in nautical miles to be:-
D = ((TAS*TAS) / tan [bank angle] ) / 34,400).
(Airspeed in knots).
So for 90 knots and 30 degrees bank, the turn diameter will be 0.408nm (or 755 metres).
Shave 20 knots off to 70, and it becomes 0.247nm or 457m
Add 20 knots on to 110, and it becomes 0.609nm, or 1,128m.
At 90 knots, if you can stay level, increasing bank angle to 45 degrees takes the turn diameter to 0.235nm or 436m. Increase to 60 degrees and you get 0.136nm or 251m.
You can play around with numbers to try and optimise it, but I think in general terms as slow as you're comfortable, then as steep a bank as you're comfortable would be about right.
I'd need to play with numbers rather more, but suspect that a more detailed answer would say around best range speed (which is around min drag) and then pull to the buffet (or the steepest that loses no height) in a steep turn at full throttle wouldn't be too far wide of the mark.
G
I would never doubt that for a level turn one needs to go slowest and at the steepest bank which doesn't stall.
David Rodgers worked out as much for executing a turnback on an EFATO.
What is more interesting is that there are two ways to reduce the turn radius as projected onto the ground below: climb or descend.
The climb case has already been covered, and has the advantage is that you can't hit the sky 
The descent case is obviously more tricky (due to the ground below) but if you have no speed (kinetic energy) to trade, and can't do a level turn, it is your only option. A very tight turn should be possible, with plenty of downward acceleration to build up speed while keeping the wing loading very low. One does a mini version of that when flying a tight base turn to land.
A few years ago I found myself in Canada, and fancied walking up a small slope to see a glacier from close-up. I guesstimated that it would take me about half an hour, and I left my friends eating lunch by the side of a lake.
Half an hour later, finding myself climbing rather than scrambling and realising that my companions had become too small to see, I did the sensible thing and turned back. When we looked at the map, the glacier was about a kilometre higher than the starting point - perhaps five times higher than I'd realised. I think part of it was that the trees were much higher than the British forests that I'm used to, and there was little else familiar in the terrain to give a sense of scale.
This is what I'm getting at when I say I find it hard to relate a distance (e.g. 500 metre turning diameter) to the terrain. I guess the real question is, when do you decide to make the turn? Because if you're forced into it you have to just do your best, whatever the circumstances, though I suppose there may be circumstances where you had alternative undesirable but possibly safer options (ditching, flying into a shallow slope etc...)
I remember having done the numbers a while ago, and the tightest turn was NOT achieved with the slowest speed. Reason is that at the slowest speed possible (Vs) you cannot bank at all and thus cannot turn. The faster you go, the more you can load the wings before they stall. And the more you can load the wings, the steeper and tighter you can turn. The limiting factor becomes the maximum wing loading for your specific aircraft and configuration.
The best turn radius I could come up with was either with flaps deployed, at Vfe, doing a 60-degree banked turn at 2G, or a turn with flaps up, at Va, 75 degrees of bank at 4G. Those numbers were within 10% of each other, and as I remember correctly, in the 200-300m radius range. Maybe even less.
Both were for a normal category aircraft, and assuming level turns. If you throw utility/aerobatics aircraft and (semi-)aerobatics turns into the mix you could go tighter still. On the other hand - steep turns should be relatively familiar to a PPL pilot. To pull off a hammerhead (or stall turn as it's known in the UK) in a non-aerobatics aircraft, under a low cloud base, close to the ground, with no clear horizon as reference and without a lot of aerobatics experience, is a recipe for disaster.
I don't have time to re-do the calculations right now. They must be on PPRuNe somewhere.
To add something that I thought of on my way to the supermarket.
First, so people don't get the wrong impression, Ghengis' formula is not wrong. For any given angle of bank, the tightest turn is achieved with the lowest speed that allows you to fly at that bank angle. Ghengis' formula just doesn't take into account aircraft limitations like stall speed and structural load limits. You can plug in numbers that lead to a very low radius, but which the aircraft cannot fly. (Try plugging in a turn flown at 10 knots and 80 degree AoB for instance.)
Second, the tightest turns require steep turns with bank angles that are well outside the PPL exam requirements. If you are not happy with 60 degree or more bank angles, the tightest turn is achieved by flying at the steepest bank angle that you are prepared to fly, and with the lowest speed that allows you to sustain that bank angle without stalling. But in that case you are not using the full capabilities of the aircraft, and your turn radius will suffer.
Third, the steepest turn with flaps down, assuming a 2G structural limit (60 degree AoB), is not flown at Vfe, but at 1.4Vs0. Provided of course that 1.4Vs0 is lower than Vfe.
Standby one!
However brief comments before I pop back-
(1) Published research on the turnback shows 45 degree angle of bank is optimal. However, that's based upon height loss per angular change, NOT on minimum radius of turn.
(2) Any aircraft with a 2g structural limit should not be flying. The minimum structural limit for a light aeroplane is 3.8g. In a level turn you'll hit that at 75 degrees of bank.
(3) The maximum permitted stalling speed (flaps down) for a single engined light aeroplane is 61 knots.
Back shortly...
G
Re tight turns: my father saved his bacon by doing a steep turn on the upwind from 500 feet or so, and landing from where he'd come... There were two things that made it successful: his understanding that a steep angle of bank generates the smallest calculated loss of height and the aircraft being a canard. He says he stalled the front wing unintentionally, and now (30 years later) says that he thinks he'd have spun in trying it in a conventional aircraft due to lack of skill and practice. I try to bear that in mind!
