Signed contract with new employer but want to stay with current employer

Go back and tell the new employer and see what they say, if they want you they might increase your offer. You're in a good position.

Decide where you REALLY want to work first.

And be aware of the reasons you decided to leave in the first place. If those are not addressed, is the salary increase worth it?

There was some stat thrown out on a course I did years ago that 9/10 people leave anyway a few months down the line if the reasons are not addressed. I.e. as a company you are throwing good money after bad.

Doesn't really matter what they want. What do you want?

Don't worry about the legal implications.

You're on probation for the first six months: the new employer can let you go with no notice (and will if it suits them).

And you can let them go too.

It's increasingly common for people to sign up but not actually start.

I'd also caution that although your current employer is offering you increased pay, they're only doing that because it's the only way they can try to stop you leaving. They could have done it any time before now if they'd wanted. They may feel that they're giving you a more-than-adequate raise now (beyond what they would otherwise have given), so future raises might be very underwhelming.

Unless it is a very senior position the new employer will not do anything. What could they do realistically? There is no gain in it for then.

None. Look at it the other way, the new employer could let you go in the first 6 months and you wouldn't have any rights. Common enough for contracts not to proceed. We have an employee who left us 3 week's ago back with us having ditched the new job