I don't think that a nanny for one day a week is worthwhile. More hassle than it's worth... good though if you didn't have the help of Mum and MIL, as a full-time nanny can be a lower cost option with 3 or more children, than paying for childminders/nursery.
A nanny could also cost you £100 for a day... possibly more. Whereas you are only really needing care for your 2 year old, which a nursery could do and would not cost £100 a day. Trouble is finding a nursery that will do just one day.
Many nursery websites don't list fees... found one which does Thorley Day Nursery, Timperley £44 a day for a 2 year old... minimum of 2 days.
So those two days at nursery, would in my view be cheaper than having a nanny for 1 day... and less hassle. Though you don't get the children's washing done! If your employer provides Childcare Vouchers, they could be used to pay the nursery fees.
So for now, I would say keep looking at nurseries but look for 2 days rather than one. May find the nurseries are more interested in that.
A childminder or nursery which will do just the one day would be lowest cost... but as you say, you can't find anyone interested in doing that.
If you can find a local nursery that can do two days... Thursday and any other day... then you can juggle the care provided by MIL, Mum and Friend to cover the other 3 days.