Your broadband speed is going to limit some options here, unless you're prepared to set up a router that will do QoS properly to ensure that there is protected bandwidth available for VoIP.
My first suggestion would have been to port your numbers to a VoIP provider and bring them in over IP, retaining one or two POTS lines as a backup, but I think with only 6.5Mb/s download, and far less upload, this isn't going to work. I'm guessing that you are on ADSL and aren't likely to get VDSL or Virgin for a very long time, if at all?
So you need to convert your two POTS lines to IP using one or more devices with FXO ports. A quick eBay search finds this which looks fine for the job. Either you can connect your IP phones to the SIP trunks produced by the above device, or for more features, you can add a PC of some description - a Raspberry Pi will be fine. This will need to run Asterisk, and probably FreePBX - a GUI front-end to administer it. This will be essential if you want to be able to transfer calls between phones, and make simple calls between extensions, amongst other things. There are various places offering prebuilt IP PBXs based on Asterisk. I have always rolled my own as then I have control and some reassurance that it's not locked down, or 'dialling home'.
You can then take your pick of VoIP phones. I like Snom, but pretty much any SIP phone will be fine. PoE is very much a feature of most manufacturers products. I much prefer a phone with a decent web interface to manage the settings and phonebook etc, but most are good for that kind of thing these days. Yealink is slightly less well-rounded than Snom, but still perfectly reliable and I know several offices running on them without issue. Beware of routinely using the built-in switch on any phone to routinely plug in a PC etc. Not all of them are gigabit, also, if you reboot the phone for a config change etc, it usually reboots the switch too, which can be annoying.
I've just looked around a little, and it seems that there are some more options for SIP/DECT handsets. I have no experience of these, so can't offer a personal opinion here. Most of them look far more acceptable than the majority of wired phones though, given some at least will be in the domestic setting.
You can connect POTS phones using an ATA to turn the SIP into an FXS port. Useful for a fax machine or Sky box. Also if you already have a phone(s) that you want to keep using.
Alternatively, Draytek make some nice routers which have built-in VoIP capability - the fully fledged IPPBX is pricey, but does keep things simple, and relieves you of several hardware boxes and the complexity of Asterisk. I am not aware of any other manufacturers which do anything similar suitable for home use.
Whatever you do, please make sure that you use strong SIP passwords. A hacked SIP account used to make continuous multiple calls to a Russian premium-rate number can be very costly!
If you need a VoIP (or broadband) supplier, I can highly recommend Andrews & Arnold as a satisfied customer.
I hope this is of some help, and that I am interpreting your answers above correctly.
Chris