i don't think so, there is so much fighting in the bbc over where the money goes, what with new bbc tv channels also wanting more money - i don't see where the investment is going to come from, as far as the execs at the bbc are concerned, the web is low status.
A successful bbc search engine could damage uk e-commerce significantly as it would not have to have ads on it - hence folk may well prefer to use it over commercial rivals. The knock on effect would mean that (over a year) millions of uk customers would not be clicking on adverts from the uks wide variety of e-commerce websites - hurting profits and costing jobs.
...unless the beeb is allowed to have ads on it.
In my opinion a bbc search engine would have to offer something that the other search engines can't provide to justify its existence (as we're paying for the bbc) - otherwise why should we pay for something that others (google etc) already do fine? I have no idea what they could offer that others don't (then again bbc news is arguable the worlds best news site, so perhaps I shouldn't be quick to judge).
The beeb are also actively looking to cut back on their websites just now, so it seems strange to start looking into producing a search engine.