Welcome to the Forum.
There is a great tool on the Unlock Information Hub to "Establishing what level of check an employer can carry out" and I attach a link https://hub.unlock.org.uk/knowledgebase/establishing-eligibility/.
Just because you have convictions does not mean that the employer will revoke the job offer. You have clearly shown that you are the best person for the job and that you have the necessary skills and experience. The convictions are over 9 years old and you have explained the circumstances surrounding the convictions to HR. It's hard to know what this new employer will do but, I would have thought that if they intended to revoke the offer then they would have done this before you started work on Monday. It is more than likely that they may want to have a discussion with you when you arrive on Monday, so be prepared for this.
I understand why you haven't telephoned them but, this may be the best thing to do in order to set your mind at rest.
If you feel that it would be useful to have a chat about
this, please feel free to give the Unlock Helpline a call.
Unlock Helpline - confidential peer advice on overcoming the effects of criminal convictions
Call: 01634 247350
Text: 07824 113848
Write: Helpline, Unlock, Maidstone Community Support Centre, 39-48 Marsham Street, Maidstone, Kent, ME6 5AG
For online self-information information, visit hub.unlock.org.uk
Need Unlocks advice? Visit our self-help information site
or contact our helpline