According to Reuben Fine in "Basic Chess Endings," the "longest mates" (on an 8x8 board) are as follows:

Pieces.................Longest Mate
----------.................--------------------
K+Q vs K.................10 moves
K+R vs K.................17 moves
K+B+B vs K.............18 moves
K+B+N vs K.............34 moves

All but the K+B+N vs K should be essentially the same on a 10x8 board, where any extra number of required moves would be small and due only to the extra space on a 10x8 board. The K+B+N vs K on a 10x8 board is more interesting, as Ed has pointed out. Once the lone King has been forced to a corner opposite in color from the Bishop, the lone King must be forced to the closer corner of the same color as the Bishop, otherwise the corralling procedure will fail. I haven't studied this enough to know for sure if it can be forced at all (although I suspect Ed has worked it out). The K+C vs K is easy since a Chancellor can be used as a "supercharged Rook." The K+A vs K is not really hard, but the procedure certainly has no counterpart in regular 8x8 chess, so seems strange when an 8x8 chessplayer first tries it.

I'm not sure I understand why Ed thinks the Lucena position needs a complete rewriting on a 10x8 board. The winning procedure is independent of the "width" of the board and would be the same on a 20x8 board. I suspect that many, if not most, endings do not really depend a great deal on the "width" of the board (the K+B+N vs K ending being a notable exception); the "height" of the board is the determining factor. I freely admit that I don't have a proof of this assertion, but it would surprise me if 8x8 endgame theory changed significantly on a 10x8 board. For those who are interested, a more dramatic geometric effect is produced by using an Omega Chess board (10x10 with 4 corner squares). On this board, several strange things happen:
1. K+R can no longer force mate on a lone K.
2. K+B+N can no longer force mate on a lone K.
3. An _unassisted_ Queen can force mate on a lone King.
4. K+N+N _can_ force mate on a lone K.
Omega Chess turns out to be a much slower game than Gothic Chess, and I'll leave it at that, since this is a Gothic Chess discussion board (and Gothic Chess is more exciting anyway).