Are these all-cotton bed linens? All white or colored? With or without lace, pulled worked, tatting, etc.?

If all-cotton, white and without fragile let-in trim or lace, then wash at relatively warm/high temps. (I regularly wash mine at 150+F.) Cotton is pretty tough. It is damaged by prolonged exposure to moisture, though nothing ordinary, such as spot-removing long soaks, would qualify as dangerously-prolonged exposure. (If you ever have a mind-lapse and soak far too long, i.e. many days/weeks, then gently remove stuff from water and let the fabrics line-dry to full dry before continuing to launder.) I use high temps in the wash to eliminate the need for bleaches (of whatever sort).

As for drying, I line dry all bed linen, the ulitmate in "cold drying". If I chose to machine dry, I certainly wouldn't use cold temps, or even cool ones, for cotton. The extra abrasion from long drying times necessitated by cold temps would be harder on the fabric than medium, medium-hot temps and shorter drying times.

Shrinkage with cottons is always a possibility, but it's not cumulative, once it's shrunk, it's shrunk, in my experience. And it's usually minimal, especially with high quality, long-staple cottons. Elastics on fitted sheets can sometimes be damaged by high temps, but that is easier to remediate than fabric which has overall damaged from too-liberal use of bleaching and cleaning agents necessitated by washing in cooler temps.

Now if you have linens that are colored, then lower wash temps might be in order. (But not drying temps as the issue is hotter water releasing dye into the water, which doesn't happen in the dryer.)

If your linens have fancy lace, tatting, or pulled work, temps are not as much as issue as mechanical strain and abrasion problems, so I'd opt for gentler action, shorter wash periods, and of course, line drying (carefully pinned so as not strain the more-tender components, especially when wet.) I would assume that if your stuff is highly ornamented, it's not for every day use, so you might find that as added insurance during laundering, that it's worth your while to baste sections of mesh, cambric or percale over spans of the trim to support it during each wash. This I would only do when the material required almost a fabric conservator's care, and thus not in use very often.

As for soap, use whatever works well in your water for other stuff, and in general less, rather than more, as long as you're getting good cleaning. There are issues regarding OBAs, or optical brightening agents (either applied to fabric in the bolt, or added to washing powder to replace what's lost during wash.) You can read about these issues here, but as I don't know the age of your fabrics (really old stuff didn't have it applied during manufacture), what's common in the UK/EU markets for white fabrics as it is in the US, and whether EU/UK detergents routinely have OBAs in the wash powder. In general, if your stuff had OBAs applied during manucfacture you may experience some blotchiness if you don't continually reapply the OBAs removed by use and washing. Conversely, if your stuff is orginally OBA-free, then use of OBAs can cause blotchiness. I have a mixed bag here, mostly old white OBA-free cotton and linen sheets and table linens, so I use an OBA free powder. I do have some modern fabrics, and frankly, I don't see big problems with blotchiness from washed-out OBAs, even without using an OBA-added powder. OBAs are actually just weakly-persistent flourescent dyes that make white fabrics appear whiter to the eye, but I prefer not to use them.

As always, though: YMMV (US-speak for "your mileage may vary", i.e., what works for me, may not work for you.) But I hope this helps, nonetheless.

L