Seeing a lot of people recommending iZettle and other smartphone devices, OP specified they didn't want these and asked for a Wifi or Phone line based terminal recommendations...
I would recommend following up the offer of going with Worldpay (formerly Streamline) and getting a Static Desktop terminal on IP (broadband). I would expect you to pay £15-20 a month. Debit and credit quotes are now % only (not pence for debit) so haggle as low as possible. Depending on the business' ATV you should try aim for c.0.5% or less to keep txn charges to a minimum.
Also check contract lengths (18m with Worldpay I believe) and also any minimum monthly service charge, try negotiate no MMSC if txn volumes are not going to be very high.
Agreed something like this. Talk with your bank normally they have a preferred card merchant or provided merchant services themselves. I'm with HSBC both business and current account and received favorable rates.