If you care about your old employer's impression of you, your best bet is simply explaining to your new employer that you made a mistake, and say you will need to give two week's notice.
If they work office hours, do this by email ASAP and, if you've not heard back, follow up with a 'phone call first thing Monday, before you speak to your old employer.
If they didn't give an indication that timing was a major factor in their decision-making process (i.e. it's peak season), they will probably be understanding and won't hold it against you (unless your new job is proofreading contracts!). Most employers are far more willing to wait a week than call up second preferences, who would then have to decide, give a week, etc.
As to your current employer, there is no 'good' way of telling them you don't want to work your notice period.
You could ask them... but it's generally not a good idea to tell an employer you're leaving them unless you have a job to go to that you can 100% definitely take up.
If you can't change it, it is a breach of contract for you to leave without working your notice. In practice, unless your employer enjoys being petty, or there is a specific penalty clause in your contract, they are unlikely to pursue you legally (NB: none of this is advice in the legal/employment sense: rely on your union or solicitor for that).
If you have a two week notice period, it is typically because your employer is just giving you the legal minimum based on your length of service and asking for the same in return. Two weeks is not enough to hire a replacement, it's a window for knowledge transfer at best. One week won't make much difference, so they may not even care.
- They may mention this in any future reference. (Check: is your offer conditional or 'subject to suitable references')?
- They might fire you on the spot, leaving you with a week's less pay.
- They probably won't hire you in future.
- Generally, this will leave a bad impression.
The only way I can see this ending well is if your employer is generally not very process-oriented, really likes you, and you're not going to work for a competitor. In this case, you could go to them with an approach like:
I'm really sorry to put you in this position, but I've had an offer to do X, which is a really great opportunity for me - but they need me to start next week! Would you be willing to let me go on Friday - I'm willing to do Y and Z to make this work.
However, if they do say no, they will know that you have considered leaving, and so they might start to look to replace you.
You always have the option of not taking the new job.
Your old employer needn't ever know you got a new job. If your new employer is firm about their start date, saying "It would have been lovely but I'm sorry it didn't work out on this occasion" as a parting message won't burn any bridges (and they might reconsider).