The dreaded 3 month notice policy....my last job had this. I agonised over it for ages and asked everyone I knew what to do.
I tried to negotiate with the company to leave earlier as I had already started with my new company (I worked only 3 days per week with the previous company so it was possible) but they weren't having any of it.
In the end I wrote a resignation letter and said I couldn't do two jobs at the same time effectively and that I was leaving with effect from XYZ. I'd already cleared my things in preparation. I was at least expecting a barrage of emails about how unprofessional I was (which I agree would have been well deserved) but in fact there was absolutely nothing. They totally blanked me from that point onwards. I was worried about repercussions such as letters about charging for temporary cover and so forth, but not a word.
The only thing which happened was they withheld my P45 for two months, I discovered they weren't supposed to do this so I actually contacted the tax office as, of course, I was paying basic tax until it was sorted out. The tax office (I presume) must have nudged them and it magically appeared, sent to my address, no letter or anything.
To be honest I'm glad I left, I couldn't have done three months' notice. Nothing dreadful has come out of walking away sooner. I would advise anyone to offer a date and then go, there is little they can do. You are technically in breach of contract but it would cost so much for them to chase you that, unless you are something high up like the governor of the Bank of England-type, in reality they aren't going to bother.