Have the potential of taking on a on a contract whereby I am doing two days a week (15 hours) and paid on that basis

I can choose the days of the week as appropriate, work from home or their office as I see fit (mainly going in for meetings) some client visits/meetings too where I will be their official representative and hold a title. Still using my own computer etc. If my work isn’t done in the hours I will likely continue but not receive additional payment

Ordinarily, I would think this would be an open and shut self employment case, however the specified number of hours/days and regularity, plus holding a formal title, makes me doubt myself and wonder if this shouldn’t really be treated as being a part time employee,

Am I being paranoid? maybe better to try and get them to drop any mention of hours?

Could you send someone else to do the job if you were working elsewhere? HMRC have a disguised employment questionnaire on their website. Google IR85 I think it is.

IR35? I thought that was only if you were using a shell/personal services company rather than self employed.

Good question on sending someone else, I suspect I could for most issues

Are you working through your own company or as an individual?
Your wording implies it is the latter, in which case IR35 is not relevant and the question is whether you are self-employed or employed. The latter has advantages (employment rights being the most obvious one) but comes with a higher employee rate of NIC. Having said that, the lower self employed rate was due to be increased, but that budget proposal didn’t make it to the Finance Act as a result of the election.
The key financial difference is that employed means the client/employer has to pay employer’s NIC but under self-employment they don’t. In theory that doesn’t affect you, although some companies will attempt to insert a clause into the contract passing the risk to you. Tell them to do one if they go down that road.
If you are through a company then IR35 is the issue but before we get into that (and it is horribly complicated), can you confirm if you are through a company?

Nope, individual

In which case you can relax once you have checked that any contract does not attempt to pass tax risk on to you.
The only other thing to bear in mind is whether you need to register for VAT (or, if you have VATable inputs, whether you might actually want to register).

Thanks, no, everything will be well under the vat limit and no significant materials costs etc. so no problem there – thanks for the help