The short version
In England and Wales, picking wild fungi growing on land you have lawful access to is not theft, provided you are not doing it for reward, for sale, or for any other commercial purpose. That comes from section 4(3) of the Theft Act 1968, which specifically exempts picking mushrooms, flowers, fruit and foliage growing wild on any land. The moment money is involved, the exemption falls away.
Access is a separate question from picking. The Theft Act does not give you the right to be somewhere. You still need a public right of way, open access land, or the landowner's permission.
Uprooting is different from picking
Section 13 of the Wildlife and Countryside Act 1981 makes it an offence to uproot any wild plant without the authorisation of the landowner or occupier. For fungi this matters because the fruiting body is what you take; the mycelium below stays in the ground. Cut or twist cleanly and do not rake or dig through the leaf litter.
Protected species
A small number of British fungi are listed on Schedule 8 of the Wildlife and Countryside Act 1981 and must not be picked at all. They include the Royal Bolete (Boletus regius), Bearded Tooth or Lion's Mane (Hericium erinaceus) and the Oak Polypore (Buglossoporus quercinus). Learn them so you can leave them.
Where picking is banned outright
Byelaws and site rules routinely override the general position. The New Forest has operated a no-picking policy for years. Many National Trust properties, local authority parks, nature reserves and Sites of Special Scientific Interest prohibit collecting entirely, or restrict it to a named permit holder. Epping Forest and several Royal Parks are likewise off limits.
Scotland is different again. The Land Reform (Scotland) Act 2003 gives broad access rights, and the Scottish Outdoor Access Code permits picking fungi for personal use, with the same expectation of restraint and respect for growing crops and private gardens.
The forager's code
Never take more than you will genuinely eat. A basket half full is a good day.
Leave the small ones and leave the very old ones. Old caps are spore factories and young ones are tomorrow's dinner.
Use an open basket or a net bag so spores drop as you walk.
Cut cleanly at the base or twist out and backfill the hole. Never rake the litter layer.
Take no more than a third of what is fruiting in any one patch, and move on.
Ask the landowner. A polite conversation buys years of access; being caught unasked ends it permanently.
Take nothing you have not identified with complete certainty, and never eat a wild mushroom on the strength of a single source.