If you are an attention seeker, you came to the right place, Pavel.
First, congratulations on your decision. I know you and Westie will be happy and will be OK financially.
As for your English, I think it is excellent. Most native English speakers don't write as well as you. I realize that spoken English is different. There's nothing like immersing yourself in the culture, speaking and hearing the language every day to improve your skills. I think you will lose much of the Czech accent with practice and will soon have no problem being understood. My experience with the British is that they have numerous native accents and some of them are difficult to understand. I would think employers are accustomed to that and to making accomodations for accents.
The most important thing you can do in preparing for the job hunt is to change your attitude about yourself. Don't underestimate yourself. Concentrate on your strengths. You will bring a fresh outlook to the workplace, something good employers will recognize as an asset to their company. Don't dwell on being a foreigner; you are a European, just like the person on the other side of the desk. One thing to do in identifying potential employers is to see if there are any with business interests in other European countries. Do you speak more than two languages? I know you want to live in London and would obviously prefer not being away very often, but there may be a real gem of an opportunity available.
Perfectionism can be a problem. I know from my own early work experience it can be a real stumbling block, taking valuable time away from more productive work. We sometimes strive for perfection because we are afraid of being wrong. Mistakes are the greatest teachers; don't be afraid of them.
I don't know what your profession is, so I can't give any advice beyond the rather general comments above. I do wish you all the best in your job search.