Unfortunately the UC regs do not require the jobcentre to excuse you from work-related activity for anything more than a temporary period with a sick note (unlike ESA, where the sick note excused you until you were assessed by the benefit agency).
You need to consider two things here (1) whether you should go back and challenge the ESA refusal; and (2) how you can get UC to play ball.
Firstly the ESA then - the initial stage of challenging the decision is by Mandatory Reconsideration (i.e. you ask the DWP to look at their own decision again). The normal deadline for this is one month, but you have up to 13 months to request this if you have a good explanation why (e.g. physical/mental health problems mean that you find it difficult to act promptly on such things than others would). Once you have the Mandatory Reconsideration you can appeal to a Tribunal on form SSCS1 (google it).
Three instances this may not work for you (1) if you're in a 'full service' area i.e. your UC claim is online, then there's no going back I'm afraid; (2) if you're outside the 13 month absolute deadline; or (3) you want to stick with UC now you're on it. If any of these apply then you need to consider what measures you can take to make UC work better for you. If you feel you are not fit to look for work, then you need to make sure you are in the process of being assessed for Limited Capability for Work - this works in the same way as ESA i.e. you need to complete a UC50 and then go for a medical - however until you have been to that medical and been deemed unfit for work, then you will not be excused from jobsearch etc except at the discretion of your Work Coach - so try to get them on side if you can.
If you feel you can look for work, or while you are in the process of being assessed, do not forget that as a public body, the jobcentre have an obligation to make reasonable adjustments in order to facilitate your disabilities in accordance with the Equalities Act 2010 - this may mean dealing with you over the phone, or restricting the type of work which would be suitable for you etc.
Hope this helps, but do let us know if you need anything further.