Welcome to Flyertalk and welcome to the BA forum retfalvid, it's great to see you here and I hope we will see more from you in the future.

If it was a fully online booking then normally it would ticket immediately, and the e-ticket would follow in typically 4 to 12 hours. That said, the PNR is usually very clearly shown on the final screen of the booking process, so if you didn't get that then I'm not sure your booking went through. I don't think email is a good way to progress this, so probably best to call BA and see if they can see whether the booking has taken. Note that you have 24 hours to fully cancel your booking if there was a mistake on it, so you may need to move quickly here, depending on what you mean by "recently". The other option is to use Twitter to send a DM to BA, someone will typically call you back. Agents can search for someone by flight number and surname, but they are reluctant to get PNRs out, for obvious security reasons, but may be able to check the ticketing status to you.

When I book, my credit card tends to show the transaction within 2 or 3 days.